% Mizar ND problem: t39_jordan2c,jordan2c,939,42 fof(dh_c1_46__jordan2c,definition, ( ( m2_subset_1(c1_46__jordan2c,k1_numbers,k5_numbers) => ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k14_euclid(c1_46__jordan2c)))) => ~ ( r1_xreal_0(1,c1_46__jordan2c) & A = k1_euclid(c1_46__jordan2c) & v6_tbsp_1(A,k14_euclid(c1_46__jordan2c)) ) ) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k14_euclid(B)))) => ~ ( r1_xreal_0(1,B) & C = k1_euclid(B) & v6_tbsp_1(C,k14_euclid(B)) ) ) ) ), introduced(definition,[new_symbol(c1_46__jordan2c),file(jordan2c,c1_46__jordan2c)]), [interesting(0.8),axiom,file(jordan2c,c1_46__jordan2c)]). fof(dh_c2_46__jordan2c,definition, ( ( m1_subset_1(c2_46__jordan2c,k1_zfmisc_1(u1_struct_0(k14_euclid(c1_46__jordan2c)))) => ~ ( r1_xreal_0(1,c1_46__jordan2c) & c2_46__jordan2c = k1_euclid(c1_46__jordan2c) & v6_tbsp_1(c2_46__jordan2c,k14_euclid(c1_46__jordan2c)) ) ) => ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k14_euclid(c1_46__jordan2c)))) => ~ ( r1_xreal_0(1,c1_46__jordan2c) & A = k1_euclid(c1_46__jordan2c) & v6_tbsp_1(A,k14_euclid(c1_46__jordan2c)) ) ) ), introduced(definition,[new_symbol(c2_46__jordan2c),file(jordan2c,c2_46__jordan2c)]), [interesting(0.8),axiom,file(jordan2c,c2_46__jordan2c)]). fof(e1_46__jordan2c,assumption, ( r1_xreal_0(1,c1_46__jordan2c) & c2_46__jordan2c = k1_euclid(c1_46__jordan2c) ), introduced(assumption,[file(jordan2c,e1_46__jordan2c)]), [interesting(0.8),axiom,file(jordan2c,e1_46__jordan2c)]). fof(e2_46__jordan2c,assumption,( v6_tbsp_1(c2_46__jordan2c,k14_euclid(c1_46__jordan2c)) ), introduced(assumption,[file(jordan2c,e2_46__jordan2c)]), [interesting(0.8),axiom,file(jordan2c,e2_46__jordan2c)]). fof(reflexivity_r1_tarski,theorem,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), [interesting(0.9),axiom,file(tarski,r1_tarski)]). fof(cc1_arytm_3,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_ordinal1(B) & v2_ordinal1(B) & v3_ordinal1(B) ) ) ) ), file(arytm_3,cc1_arytm_3), [interesting(0.9),axiom,file(arytm_3,cc1_arytm_3)]). fof(cc1_finseq_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) ) ), file(finseq_1,cc1_finseq_1), [interesting(0.9),axiom,file(finseq_1,cc1_finseq_1)]). fof(cc1_xreal_0,theorem,( ! [A] : ( v4_ordinal2(A) => v1_xreal_0(A) ) ), file(xreal_0,cc1_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc1_xreal_0)]). fof(cc2_arytm_3,theorem,( ! [A] : ( ( v1_xboole_0(A) & v3_ordinal1(A) ) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ) ), file(arytm_3,cc2_arytm_3), [interesting(0.9),axiom,file(arytm_3,cc2_arytm_3)]). fof(cc2_funct_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ) ), file(funct_1,cc2_funct_1), [interesting(0.9),axiom,file(funct_1,cc2_funct_1)]). fof(rc1_arytm_3,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ), file(arytm_3,rc1_arytm_3), [interesting(0.9),axiom,file(arytm_3,rc1_arytm_3)]). fof(rc1_finseq_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc1_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc1_finseq_1)]). fof(rc1_funct_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc1_funct_1), [interesting(0.9),axiom,file(funct_1,rc1_funct_1)]). fof(rc2_finset_1,theorem,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v2_funct_1(B) & v1_ordinal1(B) & v2_ordinal1(B) & v3_ordinal1(B) & v4_ordinal2(B) & v1_finset_1(B) ) ), file(finset_1,rc2_finset_1), [interesting(0.9),axiom,file(finset_1,rc2_finset_1)]). fof(rc2_funct_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) ), file(funct_1,rc2_funct_1), [interesting(0.9),axiom,file(funct_1,rc2_funct_1)]). fof(rc3_finseq_1,theorem,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc3_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc3_finseq_1)]). fof(rc3_funct_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ), file(funct_1,rc3_funct_1), [interesting(0.9),axiom,file(funct_1,rc3_funct_1)]). fof(rc4_funct_1,theorem,( ? [A] : ( v1_relat_1(A) & v3_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc4_funct_1), [interesting(0.9),axiom,file(funct_1,rc4_funct_1)]). fof(rc6_finseq_1,theorem,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) ) ), file(finseq_1,rc6_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc6_finseq_1)]). fof(rc7_finseq_1,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) ), file(finseq_1,rc7_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc7_finseq_1)]). fof(rc8_finseq_1,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc8_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc8_finseq_1)]). fof(antisymmetry_r2_hidden,theorem,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), [interesting(0.9),axiom,file(hidden,r2_hidden)]). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]). fof(dt_k1_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,k1_zfmisc_1)]). fof(dt_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), [interesting(0.9),axiom,file(ordinal2,k5_ordinal2)]). fof(cc10_membered,theorem,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,A) => v1_xcmplx_0(B) ) ) ), file(membered,cc10_membered), [interesting(0.9),axiom,file(membered,cc10_membered)]). fof(cc11_membered,theorem,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) ) ) ) ), file(membered,cc11_membered), [interesting(0.9),axiom,file(membered,cc11_membered)]). fof(cc12_membered,theorem,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_rat_1(B) ) ) ) ), file(membered,cc12_membered), [interesting(0.9),axiom,file(membered,cc12_membered)]). fof(cc13_membered,theorem,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc13_membered), [interesting(0.9),axiom,file(membered,cc13_membered)]). fof(cc14_membered,theorem,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v4_ordinal2(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc14_membered), [interesting(0.9),axiom,file(membered,cc14_membered)]). fof(cc16_membered,theorem,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_membered(B) ) ) ), file(membered,cc16_membered), [interesting(0.9),axiom,file(membered,cc16_membered)]). fof(cc17_membered,theorem,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) ) ) ) ), file(membered,cc17_membered), [interesting(0.9),axiom,file(membered,cc17_membered)]). fof(cc18_membered,theorem,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) ) ) ) ), file(membered,cc18_membered), [interesting(0.9),axiom,file(membered,cc18_membered)]). fof(cc19_membered,theorem,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) ) ) ) ), file(membered,cc19_membered), [interesting(0.9),axiom,file(membered,cc19_membered)]). fof(cc1_membered,theorem,( ! [A] : ( v5_membered(A) => v4_membered(A) ) ), file(membered,cc1_membered), [interesting(0.9),axiom,file(membered,cc1_membered)]). fof(cc20_membered,theorem,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) & v5_membered(B) ) ) ) ), file(membered,cc20_membered), [interesting(0.9),axiom,file(membered,cc20_membered)]). fof(cc2_finset_1,theorem,( ! [A] : ( v1_finset_1(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_finset_1(B) ) ) ), file(finset_1,cc2_finset_1), [interesting(0.9),axiom,file(finset_1,cc2_finset_1)]). fof(cc2_membered,theorem,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), [interesting(0.9),axiom,file(membered,cc2_membered)]). fof(cc3_arytm_3,theorem,( ! [A] : ( m1_subset_1(A,k5_ordinal2) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ) ), file(arytm_3,cc3_arytm_3), [interesting(0.9),axiom,file(arytm_3,cc3_arytm_3)]). fof(cc3_membered,theorem,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), [interesting(0.9),axiom,file(membered,cc3_membered)]). fof(cc4_membered,theorem,( ! [A] : ( v2_membered(A) => v1_membered(A) ) ), file(membered,cc4_membered), [interesting(0.9),axiom,file(membered,cc4_membered)]). fof(cc4_xreal_0,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v2_xreal_0(A) ) ) ), file(xreal_0,cc4_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc4_xreal_0)]). fof(cc5_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & v3_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) ) ) ), file(xreal_0,cc5_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc5_xreal_0)]). fof(cc6_membered,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( v1_membered(A) & v2_membered(A) ) ) ), file(membered,cc6_membered), [interesting(0.9),axiom,file(membered,cc6_membered)]). fof(cc8_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(xreal_0,cc8_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc8_xreal_0)]). fof(cc9_membered,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k5_numbers)) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc9_membered), [interesting(0.9),axiom,file(membered,cc9_membered)]). fof(fc1_subset_1,theorem,( ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ), file(subset_1,fc1_subset_1), [interesting(0.9),axiom,file(subset_1,fc1_subset_1)]). fof(fc2_finseq_1,theorem, ( v1_xboole_0(k1_xboole_0) & v1_relat_1(k1_xboole_0) & v3_relat_1(k1_xboole_0) & v1_funct_1(k1_xboole_0) & v2_funct_1(k1_xboole_0) & v1_finset_1(k1_xboole_0) & v1_finseq_1(k1_xboole_0) ), file(finseq_1,fc2_finseq_1), [interesting(0.9),axiom,file(finseq_1,fc2_finseq_1)]). fof(fc5_membered,theorem, ( v1_membered(k5_ordinal2) & v2_membered(k5_ordinal2) & v3_membered(k5_ordinal2) & v4_membered(k5_ordinal2) & v5_membered(k5_ordinal2) ), file(membered,fc5_membered), [interesting(0.9),axiom,file(membered,fc5_membered)]). fof(fc6_membered,theorem, ( v1_xboole_0(k1_xboole_0) & v1_membered(k1_xboole_0) & v2_membered(k1_xboole_0) & v3_membered(k1_xboole_0) & v4_membered(k1_xboole_0) & v5_membered(k1_xboole_0) ), file(membered,fc6_membered), [interesting(0.9),axiom,file(membered,fc6_membered)]). fof(rc1_finset_1,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) ) ), file(finset_1,rc1_finset_1), [interesting(0.9),axiom,file(finset_1,rc1_finset_1)]). fof(rc1_membered,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ), file(membered,rc1_membered), [interesting(0.9),axiom,file(membered,rc1_membered)]). fof(rc1_subset_1,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) ) ) ), file(subset_1,rc1_subset_1), [interesting(0.9),axiom,file(subset_1,rc1_subset_1)]). fof(rc1_xreal_0,theorem,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(xreal_0,rc1_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc1_xreal_0)]). fof(rc2_subset_1,theorem,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) ) ), file(subset_1,rc2_subset_1), [interesting(0.9),axiom,file(subset_1,rc2_subset_1)]). fof(rc2_xreal_0,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(xreal_0,rc2_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc2_xreal_0)]). fof(rc3_finset_1,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & v1_finset_1(B) ) ) ), file(finset_1,rc3_finset_1), [interesting(0.9),axiom,file(finset_1,rc3_finset_1)]). fof(rc3_xreal_0,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & v3_xreal_0(A) ) ), file(xreal_0,rc3_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc3_xreal_0)]). fof(rc4_finset_1,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & v1_finset_1(B) ) ) ), file(finset_1,rc4_finset_1), [interesting(0.9),axiom,file(finset_1,rc4_finset_1)]). fof(rc4_xreal_0,theorem,( ? [A] : ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(xreal_0,rc4_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc4_xreal_0)]). fof(t1_subset,theorem,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), [interesting(0.9),axiom,file(subset,t1_subset)]). fof(t2_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & v3_xreal_0(B) ) => v3_xreal_0(A) ) ) ) ), file(real,t2_real), [interesting(0.9),axiom,file(real,t2_real)]). fof(t3_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( r1_xreal_0(A,B) & ~ v3_xreal_0(A) & v3_xreal_0(B) ) ) ) ), file(real,t3_real), [interesting(0.9),axiom,file(real,t3_real)]). fof(t3_subset,theorem,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), [interesting(0.9),axiom,file(subset,t3_subset)]). fof(t4_subset,theorem,( ! [A,B,C] : ( ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) ) => m1_subset_1(A,C) ) ), file(subset,t4_subset), [interesting(0.9),axiom,file(subset,t4_subset)]). fof(t5_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(B) | v3_xreal_0(A) | v2_xreal_0(B) ) ) ) ) ), file(real,t5_real), [interesting(0.9),axiom,file(real,t5_real)]). fof(t5_subset,theorem,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), [interesting(0.9),axiom,file(subset,t5_subset)]). fof(t6_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(A) | v2_xreal_0(B) | v3_xreal_0(A) ) ) ) ) ), file(real,t6_real), [interesting(0.9),axiom,file(real,t6_real)]). fof(t7_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,B) & ~ v2_xreal_0(A) & ~ v3_xreal_0(B) ) ) ) ), file(real,t7_real), [interesting(0.9),axiom,file(real,t7_real)]). fof(t8_boole,theorem,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), [interesting(0.9),axiom,file(boole,t8_boole)]). fof(t8_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,B) & ~ v3_xreal_0(B) & ~ v2_xreal_0(A) ) ) ) ), file(real,t8_real), [interesting(0.9),axiom,file(real,t8_real)]). fof(existence_m1_subset_1,axiom,( ! [A] : ? [B] : m1_subset_1(B,A) ), file(subset_1,m1_subset_1), [interesting(0.9),axiom,file(subset_1,m1_subset_1)]). fof(existence_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ? [C] : m2_subset_1(C,A,B) ) ), file(subset_1,m2_subset_1), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(redefinition_k5_numbers,definition,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), [interesting(0.9),axiom,file(numbers,k5_numbers)]). fof(redefinition_m2_subset_1,definition,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m2_subset_1(C,A,B) <=> m1_subset_1(C,B) ) ) ), file(subset_1,m2_subset_1), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), [interesting(0.9),axiom,file(numbers,k1_numbers)]). fof(dt_k5_numbers,axiom,( m1_subset_1(k5_numbers,k1_zfmisc_1(k1_numbers)) ), file(numbers,k5_numbers), [interesting(0.9),axiom,file(numbers,k5_numbers)]). fof(dt_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), [interesting(0.9),axiom,file(subset_1,m1_subset_1)]). fof(dt_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m2_subset_1(C,A,B) => m1_subset_1(C,A) ) ) ), file(subset_1,m2_subset_1), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(cc15_membered,theorem,( ! [A] : ( v1_xboole_0(A) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc15_membered), [interesting(0.9),axiom,file(membered,cc15_membered)]). fof(cc1_finset_1,theorem,( ! [A] : ( v1_xboole_0(A) => v1_finset_1(A) ) ), file(finset_1,cc1_finset_1), [interesting(0.9),axiom,file(finset_1,cc1_finset_1)]). fof(cc1_funct_1,theorem,( ! [A] : ( v1_xboole_0(A) => v1_funct_1(A) ) ), file(funct_1,cc1_funct_1), [interesting(0.9),axiom,file(funct_1,cc1_funct_1)]). fof(cc2_xreal_0,theorem,( ! [A] : ( v1_xreal_0(A) => v1_xcmplx_0(A) ) ), file(xreal_0,cc2_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc2_xreal_0)]). fof(cc3_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & v2_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc3_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc3_xreal_0)]). fof(cc6_xreal_0,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v3_xreal_0(A) ) ) ), file(xreal_0,cc6_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc6_xreal_0)]). fof(cc7_xreal_0,theorem,( ! [A] : ( ( v1_xboole_0(A) & v1_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc7_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc7_xreal_0)]). fof(fc2_membered,theorem, ( ~ v1_xboole_0(k1_numbers) & v1_membered(k1_numbers) & v2_membered(k1_numbers) ), file(membered,fc2_membered), [interesting(0.9),axiom,file(membered,fc2_membered)]). fof(t1_numerals,theorem,( m1_subset_1(0,k5_numbers) ), file(numerals,t1_numerals), [interesting(0.9),axiom,file(numerals,t1_numerals)]). fof(t1_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & v2_xreal_0(A) ) => v2_xreal_0(B) ) ) ) ), file(real,t1_real), [interesting(0.9),axiom,file(real,t1_real)]). fof(t2_subset,theorem,( ! [A,B] : ( m1_subset_1(A,B) => ( v1_xboole_0(B) | r2_hidden(A,B) ) ) ), file(subset,t2_subset), [interesting(0.9),axiom,file(subset,t2_subset)]). fof(t4_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( r1_xreal_0(A,B) & ~ v2_xreal_0(B) & v2_xreal_0(A) ) ) ) ), file(real,t4_real), [interesting(0.9),axiom,file(real,t4_real)]). fof(t6_boole,theorem,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), [interesting(0.9),axiom,file(boole,t6_boole)]). fof(t7_boole,theorem,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), [interesting(0.9),axiom,file(boole,t7_boole)]). fof(reflexivity_r1_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => r1_xreal_0(A,A) ) ), file(xreal_0,r1_xreal_0), [interesting(0.9),axiom,file(xreal_0,r1_xreal_0)]). fof(connectedness_r1_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( r1_xreal_0(A,B) | r1_xreal_0(B,A) ) ) ), file(xreal_0,r1_xreal_0), [interesting(0.9),axiom,file(xreal_0,r1_xreal_0)]). fof(rqLessOrEqual__r1_xreal_0__r0_r0,theorem,( r1_xreal_0(0,0) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_r0)]). fof(rqLessOrEqual__r1_xreal_0__r0_r1,theorem,( r1_xreal_0(0,1) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_r1)]). fof(rqLessOrEqual__r1_xreal_0__r1_r0,theorem,( ~ r1_xreal_0(1,0) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_r0)]). fof(rqLessOrEqual__r1_xreal_0__r1_r1,theorem,( r1_xreal_0(1,1) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_r1)]). fof(spc0_numerals,theorem, ( v2_xreal_0(0) & m2_subset_1(0,k1_numbers,k5_numbers) & m1_subset_1(0,k5_numbers) & m1_subset_1(0,k1_numbers) ), file(numerals,spc0_numerals), [interesting(0.9),axiom,file(numerals,spc0_numerals)]). fof(spc1_numerals,theorem, ( v2_xreal_0(1) & m2_subset_1(1,k1_numbers,k5_numbers) & m1_subset_1(1,k5_numbers) & m1_subset_1(1,k1_numbers) ), file(numerals,spc1_numerals), [interesting(0.9),axiom,file(numerals,spc1_numerals)]). fof(spc0_boole,theorem,( v1_xboole_0(0) ), file(boole,spc0_boole), [interesting(0.9),axiom,file(boole,spc0_boole)]). fof(spc1_boole,theorem,( ~ v1_xboole_0(1) ), file(boole,spc1_boole), [interesting(0.9),axiom,file(boole,spc1_boole)]). fof(fc10_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v2_xreal_0(k2_xcmplx_0(B,A)) & ~ v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc10_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc10_xreal_0)]). fof(fc11_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v2_xreal_0(k2_xcmplx_0(A,B)) & v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc11_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc11_xreal_0)]). fof(fc12_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & ~ v2_xreal_0(k2_xcmplx_0(B,A)) & v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc12_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc12_xreal_0)]). fof(fc14_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & ~ v2_xreal_0(k4_xcmplx_0(A)) ) ) ), file(xreal_0,fc14_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc14_xreal_0)]). fof(fc15_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & ~ v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc15_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc15_xreal_0)]). fof(fc16_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & ~ v2_xreal_0(k6_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc16_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc16_xreal_0)]). fof(fc19_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(A,B)) & v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & ~ v2_xreal_0(k6_xcmplx_0(A,B)) & v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc19_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc19_xreal_0)]). fof(fc20_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(B,A)) & v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & v2_xreal_0(k6_xcmplx_0(B,A)) & ~ v3_xreal_0(k6_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc20_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc20_xreal_0)]). fof(fc7_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc7_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc7_xreal_0)]). fof(fc9_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v2_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc9_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc9_xreal_0)]). fof(dt_k6_xcmplx_0,axiom,( $true ), file(xcmplx_0,k6_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k6_xcmplx_0)]). fof(fc13_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) ) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & ~ v3_xreal_0(k4_xcmplx_0(A)) ) ) ), file(xreal_0,fc13_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc13_xreal_0)]). fof(fc17_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(A,B)) & v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v2_xreal_0(k6_xcmplx_0(A,B)) & ~ v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc17_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc17_xreal_0)]). fof(fc18_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(B,A)) & v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & ~ v2_xreal_0(k6_xcmplx_0(B,A)) & v3_xreal_0(k6_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc18_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc18_xreal_0)]). fof(fc1_xreal_0,theorem,( ! [A] : ( v1_xreal_0(A) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) ) ) ), file(xreal_0,fc1_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc1_xreal_0)]). fof(fc3_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc3_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc3_xreal_0)]). fof(fc5_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc5_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc5_xreal_0)]). fof(fc8_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v2_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc8_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc8_xreal_0)]). fof(rqRealDiff__k6_xcmplx_0__r0_r0_r0,theorem,( k6_xcmplx_0(0,0) = 0 ), file(arithm,rqRealDiff__k6_xcmplx_0__r0_r0_r0), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r0_r0_r0)]). fof(rqRealDiff__k6_xcmplx_0__r0_r1_rm1,theorem,( k6_xcmplx_0(0,1) = k4_xcmplx_0(1) ), file(arithm,rqRealDiff__k6_xcmplx_0__r0_r1_rm1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r0_r1_rm1)]). fof(rqRealDiff__k6_xcmplx_0__r0_rm1_r1,theorem,( k6_xcmplx_0(0,k4_xcmplx_0(1)) = 1 ), file(arithm,rqRealDiff__k6_xcmplx_0__r0_rm1_r1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r0_rm1_r1)]). fof(rqRealDiff__k6_xcmplx_0__r1_r0_r1,theorem,( k6_xcmplx_0(1,0) = 1 ), file(arithm,rqRealDiff__k6_xcmplx_0__r1_r0_r1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r1_r0_r1)]). fof(rqRealDiff__k6_xcmplx_0__r1_r1_r0,theorem,( k6_xcmplx_0(1,1) = 0 ), file(arithm,rqRealDiff__k6_xcmplx_0__r1_r1_r0), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r1_r1_r0)]). fof(rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,theorem,( k6_xcmplx_0(k4_xcmplx_0(1),0) = k4_xcmplx_0(1) ), file(arithm,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1)]). fof(rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,theorem,( k6_xcmplx_0(k4_xcmplx_0(1),k4_xcmplx_0(1)) = 0 ), file(arithm,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0)]). fof(spc1_arithm,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k2_xcmplx_0(A,k4_xcmplx_0(B)) = k6_xcmplx_0(A,B) ) ), file(arithm,spc1_arithm), [interesting(0.9),axiom,file(arithm,spc1_arithm)]). fof(spc6_arithm,theorem,( ! [A,B,C] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) & v1_xcmplx_0(C) ) => k2_xcmplx_0(k2_xcmplx_0(A,B),C) = k2_xcmplx_0(A,k2_xcmplx_0(B,C)) ) ), file(arithm,spc6_arithm), [interesting(0.9),axiom,file(arithm,spc6_arithm)]). fof(spc8_arithm,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k2_xcmplx_0(k4_xcmplx_0(A),k4_xcmplx_0(B)) = k4_xcmplx_0(k2_xcmplx_0(A,B)) ) ), file(arithm,spc8_arithm), [interesting(0.9),axiom,file(arithm,spc8_arithm)]). fof(spc9_arithm,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k6_xcmplx_0(k4_xcmplx_0(A),k4_xcmplx_0(B)) = k6_xcmplx_0(B,A) ) ), file(arithm,spc9_arithm), [interesting(0.9),axiom,file(arithm,spc9_arithm)]). fof(t1_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k2_xcmplx_0(A,0) = A ) ), file(arithm,t1_arithm), [interesting(0.9),axiom,file(arithm,t1_arithm)]). fof(t4_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k6_xcmplx_0(A,0) = A ) ), file(arithm,t4_arithm), [interesting(0.9),axiom,file(arithm,t4_arithm)]). fof(commutativity_k2_xcmplx_0,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k2_xcmplx_0(A,B) = k2_xcmplx_0(B,A) ) ), file(xcmplx_0,k2_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k2_xcmplx_0)]). fof(involutiveness_k4_xcmplx_0,theorem,( ! [A] : ( v1_xcmplx_0(A) => k4_xcmplx_0(k4_xcmplx_0(A)) = A ) ), file(xcmplx_0,k4_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k4_xcmplx_0)]). fof(commutativity_k9_binop_2,theorem,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k9_binop_2(A,B) = k9_binop_2(B,A) ) ), file(binop_2,k9_binop_2), [interesting(0.9),axiom,file(binop_2,k9_binop_2)]). fof(redefinition_k10_binop_2,definition,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k10_binop_2(A,B) = k6_xcmplx_0(A,B) ) ), file(binop_2,k10_binop_2), [interesting(0.9),axiom,file(binop_2,k10_binop_2)]). fof(redefinition_k9_binop_2,definition,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k9_binop_2(A,B) = k2_xcmplx_0(A,B) ) ), file(binop_2,k9_binop_2), [interesting(0.9),axiom,file(binop_2,k9_binop_2)]). fof(dt_k10_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k10_binop_2(A,B),k1_numbers) ) ), file(binop_2,k10_binop_2), [interesting(0.9),axiom,file(binop_2,k10_binop_2)]). fof(dt_k2_xcmplx_0,axiom,( $true ), file(xcmplx_0,k2_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k2_xcmplx_0)]). fof(dt_k4_xcmplx_0,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_xcmplx_0(k4_xcmplx_0(A)) ) ), file(xcmplx_0,k4_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k4_xcmplx_0)]). fof(dt_k9_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k9_binop_2(A,B),k1_numbers) ) ), file(binop_2,k9_binop_2), [interesting(0.9),axiom,file(binop_2,k9_binop_2)]). fof(dh_c3_46__jordan2c,definition, ( ? [A] : ( m1_subset_1(A,k1_numbers) & ~ r1_xreal_0(A,0) & ! [B] : ( m1_subset_1(B,u1_struct_0(k14_euclid(c1_46__jordan2c))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k14_euclid(c1_46__jordan2c))) => ( ( r2_hidden(B,c2_46__jordan2c) & r2_hidden(C,c2_46__jordan2c) ) => r1_xreal_0(k4_metric_1(k14_euclid(c1_46__jordan2c),B,C),A) ) ) ) ) => ( m1_subset_1(c3_46__jordan2c,k1_numbers) & ~ r1_xreal_0(c3_46__jordan2c,0) & ! [D] : ( m1_subset_1(D,u1_struct_0(k14_euclid(c1_46__jordan2c))) => ! [E] : ( m1_subset_1(E,u1_struct_0(k14_euclid(c1_46__jordan2c))) => ( ( r2_hidden(D,c2_46__jordan2c) & r2_hidden(E,c2_46__jordan2c) ) => r1_xreal_0(k4_metric_1(k14_euclid(c1_46__jordan2c),D,E),c3_46__jordan2c) ) ) ) ) ), introduced(definition,[new_symbol(c3_46__jordan2c),file(jordan2c,c3_46__jordan2c)]), [interesting(0.8),axiom,file(jordan2c,c3_46__jordan2c)]). fof(rc2_finseq_1,theorem,( ! [A] : ? [B] : ( m1_relset_1(B,k5_numbers,A) & v1_relat_1(B) & v1_funct_1(B) & v1_finset_1(B) & v1_finseq_1(B) ) ), file(finseq_1,rc2_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc2_finseq_1)]). fof(existence_m1_finseq_2,axiom,( ! [A] : ? [B] : m1_finseq_2(B,A) ), file(finseq_2,m1_finseq_2), [interesting(0.9),axiom,file(finseq_2,m1_finseq_2)]). fof(existence_m1_relset_1,axiom,( ! [A,B] : ? [C] : m1_relset_1(C,A,B) ), file(relset_1,m1_relset_1), [interesting(0.9),axiom,file(relset_1,m1_relset_1)]). fof(existence_m2_relset_1,axiom,( ! [A,B] : ? [C] : m2_relset_1(C,A,B) ), file(relset_1,m2_relset_1), [interesting(0.9),axiom,file(relset_1,m2_relset_1)]). fof(redefinition_m2_relset_1,definition,( ! [A,B,C] : ( m2_relset_1(C,A,B) <=> m1_relset_1(C,A,B) ) ), file(relset_1,m2_relset_1), [interesting(0.9),axiom,file(relset_1,m2_relset_1)]). fof(dt_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,k2_zfmisc_1)]). fof(dt_k4_finseq_2,axiom,( ! [A,B] : ( v4_ordinal2(A) => m1_finseq_2(k4_finseq_2(A,B),B) ) ), file(finseq_2,k4_finseq_2), [interesting(0.9),axiom,file(finseq_2,k4_finseq_2)]). fof(dt_m1_finseq_2,axiom,( $true ), file(finseq_2,m1_finseq_2), [interesting(0.9),axiom,file(finseq_2,m1_finseq_2)]). fof(dt_m1_relset_1,axiom,( $true ), file(relset_1,m1_relset_1), [interesting(0.9),axiom,file(relset_1,m1_relset_1)]). fof(dt_m2_relset_1,axiom,( ! [A,B,C] : ( m2_relset_1(C,A,B) => m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ), file(relset_1,m2_relset_1), [interesting(0.9),axiom,file(relset_1,m2_relset_1)]). fof(dt_u1_metric_1,axiom,( ! [A] : ( l1_metric_1(A) => ( v1_funct_1(u1_metric_1(A)) & v1_funct_2(u1_metric_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) & m2_relset_1(u1_metric_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) ) ) ), file(metric_1,u1_metric_1), [interesting(0.9),axiom,file(metric_1,u1_metric_1)]). fof(cc1_relset_1,theorem,( ! [A,B,C] : ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) => v1_relat_1(C) ) ), file(relset_1,cc1_relset_1), [interesting(0.9),axiom,file(relset_1,cc1_relset_1)]). fof(cc2_tbsp_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v6_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v1_finset_1(B) => v6_tbsp_1(B,A) ) ) ) ), file(tbsp_1,cc2_tbsp_1), [interesting(0.9),axiom,file(tbsp_1,cc2_tbsp_1)]). fof(fc14_finset_1,theorem,( ! [A,B] : ( ( v1_finset_1(A) & v1_finset_1(B) ) => v1_finset_1(k2_zfmisc_1(A,B)) ) ), file(finset_1,fc14_finset_1), [interesting(0.9),axiom,file(finset_1,fc14_finset_1)]). fof(fc4_subset_1,theorem,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ), file(subset_1,fc4_subset_1), [interesting(0.9),axiom,file(subset_1,fc4_subset_1)]). fof(rc3_tbsp_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v6_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(B) & v1_finset_1(B) & v6_tbsp_1(B,A) ) ) ), file(tbsp_1,rc3_tbsp_1), [interesting(0.9),axiom,file(tbsp_1,rc3_tbsp_1)]). fof(free_g1_metric_1,definition,( ! [A,B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers) & m1_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) ) => ! [C,D] : ( g1_metric_1(A,B) = g1_metric_1(C,D) => ( A = C & B = D ) ) ) ), file(metric_1,g1_metric_1), [interesting(0.9),axiom,file(metric_1,g1_metric_1)]). fof(abstractness_v1_metric_1,theorem,( ! [A] : ( l1_metric_1(A) => ( v1_metric_1(A) => A = g1_metric_1(u1_struct_0(A),u1_metric_1(A)) ) ) ), file(metric_1,v1_metric_1), [interesting(0.9),axiom,file(metric_1,v1_metric_1)]). fof(existence_l1_struct_0,axiom,( ? [A] : l1_struct_0(A) ), file(struct_0,l1_struct_0), [interesting(0.9),axiom,file(struct_0,l1_struct_0)]). fof(dt_g1_metric_1,axiom,( ! [A,B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers) & m1_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) ) => ( v1_metric_1(g1_metric_1(A,B)) & l1_metric_1(g1_metric_1(A,B)) ) ) ), file(metric_1,g1_metric_1), [interesting(0.9),axiom,file(metric_1,g1_metric_1)]). fof(dt_k13_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_funct_1(k13_euclid(A)) & v1_funct_2(k13_euclid(A),k2_zfmisc_1(k1_euclid(A),k1_euclid(A)),k1_numbers) & m2_relset_1(k13_euclid(A),k2_zfmisc_1(k1_euclid(A),k1_euclid(A)),k1_numbers) ) ) ), file(euclid,k13_euclid), [interesting(0.9),axiom,file(euclid,k13_euclid)]). fof(dt_k1_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( ~ v1_xboole_0(k1_euclid(A)) & m1_finseq_2(k1_euclid(A),k1_numbers) ) ) ), file(euclid,k1_euclid), [interesting(0.9),axiom,file(euclid,k1_euclid)]). fof(dt_l1_struct_0,axiom,( $true ), file(struct_0,l1_struct_0), [interesting(0.9),axiom,file(struct_0,l1_struct_0)]). fof(fc1_euclid,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( ~ v3_struct_0(k14_euclid(A)) & v1_metric_1(k14_euclid(A)) & v6_metric_1(k14_euclid(A)) & v7_metric_1(k14_euclid(A)) & v8_metric_1(k14_euclid(A)) & v9_metric_1(k14_euclid(A)) ) ) ), file(euclid,fc1_euclid), [interesting(0.9),axiom,file(euclid,fc1_euclid)]). fof(fc1_struct_0,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => ~ v1_xboole_0(u1_struct_0(A)) ) ), file(struct_0,fc1_struct_0), [interesting(0.9),axiom,file(struct_0,fc1_struct_0)]). fof(rc1_metric_1,theorem,( ? [A] : ( l1_metric_1(A) & v1_metric_1(A) ) ), file(metric_1,rc1_metric_1), [interesting(0.9),axiom,file(metric_1,rc1_metric_1)]). fof(rc2_metric_1,theorem,( ? [A] : ( l1_metric_1(A) & ~ v3_struct_0(A) & v1_metric_1(A) ) ), file(metric_1,rc2_metric_1), [interesting(0.9),axiom,file(metric_1,rc2_metric_1)]). fof(rc3_metric_1,theorem,( ? [A] : ( l1_metric_1(A) & ~ v3_struct_0(A) & v1_metric_1(A) & v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) ) ), file(metric_1,rc3_metric_1), [interesting(0.9),axiom,file(metric_1,rc3_metric_1)]). fof(rc3_struct_0,theorem,( ? [A] : ( l1_struct_0(A) & ~ v3_struct_0(A) ) ), file(struct_0,rc3_struct_0), [interesting(0.9),axiom,file(struct_0,rc3_struct_0)]). fof(rc5_struct_0,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(B) ) ) ), file(struct_0,rc5_struct_0), [interesting(0.9),axiom,file(struct_0,rc5_struct_0)]). fof(d1_euclid,definition,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k1_euclid(A) = k4_finseq_2(A,k1_numbers) ) ), file(euclid,d1_euclid), [interesting(0.9),axiom,file(euclid,d1_euclid)]). fof(commutativity_k4_metric_1,theorem,( ! [A,B,C] : ( ( v8_metric_1(A) & l1_metric_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => k4_metric_1(A,B,C) = k4_metric_1(A,C,B) ) ), file(metric_1,k4_metric_1), [interesting(0.9),axiom,file(metric_1,k4_metric_1)]). fof(existence_l1_metric_1,axiom,( ? [A] : l1_metric_1(A) ), file(metric_1,l1_metric_1), [interesting(0.9),axiom,file(metric_1,l1_metric_1)]). fof(redefinition_k4_metric_1,definition,( ! [A,B,C] : ( ( v8_metric_1(A) & l1_metric_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => k4_metric_1(A,B,C) = k2_metric_1(A,B,C) ) ), file(metric_1,k4_metric_1), [interesting(0.9),axiom,file(metric_1,k4_metric_1)]). fof(dt_k14_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_metric_1(k14_euclid(A)) & v6_metric_1(k14_euclid(A)) & v7_metric_1(k14_euclid(A)) & v8_metric_1(k14_euclid(A)) & v9_metric_1(k14_euclid(A)) & l1_metric_1(k14_euclid(A)) ) ) ), file(euclid,k14_euclid), [interesting(0.9),axiom,file(euclid,k14_euclid)]). fof(dt_k2_metric_1,axiom,( ! [A,B,C] : ( ( l1_metric_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k2_metric_1(A,B,C),k1_numbers) ) ), file(metric_1,k2_metric_1), [interesting(0.9),axiom,file(metric_1,k2_metric_1)]). fof(dt_k4_metric_1,axiom,( ! [A,B,C] : ( ( v8_metric_1(A) & l1_metric_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k4_metric_1(A,B,C),k1_numbers) ) ), file(metric_1,k4_metric_1), [interesting(0.9),axiom,file(metric_1,k4_metric_1)]). fof(dt_l1_metric_1,axiom,( ! [A] : ( l1_metric_1(A) => l1_struct_0(A) ) ), file(metric_1,l1_metric_1), [interesting(0.9),axiom,file(metric_1,l1_metric_1)]). fof(dt_u1_struct_0,axiom,( $true ), file(struct_0,u1_struct_0), [interesting(0.9),axiom,file(struct_0,u1_struct_0)]). fof(dt_c1_46__jordan2c,assumption,( m2_subset_1(c1_46__jordan2c,k1_numbers,k5_numbers) ), introduced(assumption,[file(jordan2c,c1_46__jordan2c)]), [interesting(0.8),axiom,file(jordan2c,c1_46__jordan2c)]). fof(dt_c2_46__jordan2c,assumption,( m1_subset_1(c2_46__jordan2c,k1_zfmisc_1(u1_struct_0(k14_euclid(c1_46__jordan2c)))) ), introduced(assumption,[file(jordan2c,c2_46__jordan2c)]), [interesting(0.8),axiom,file(jordan2c,c2_46__jordan2c)]). fof(rc1_jordan2c,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l1_metric_1(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v6_tbsp_1(B,A) ) ) ), file(jordan2c,rc1_jordan2c), [interesting(0.9),axiom,file(jordan2c,rc1_jordan2c)]). fof(rc2_tbsp_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l1_metric_1(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v6_tbsp_1(B,A) ) ) ), file(tbsp_1,rc2_tbsp_1), [interesting(0.9),axiom,file(tbsp_1,rc2_tbsp_1)]). fof(d7_euclid,definition,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k14_euclid(A) = g1_metric_1(k1_euclid(A),k13_euclid(A)) ) ), file(euclid,d7_euclid), [interesting(0.9),axiom,file(euclid,d7_euclid)]). fof(d9_tbsp_1,definition,( ! [A] : ( ( ~ v3_struct_0(A) & l1_metric_1(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v6_tbsp_1(B,A) <=> ? [C] : ( m1_subset_1(C,k1_numbers) & ~ r1_xreal_0(C,0) & ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ! [E] : ( m1_subset_1(E,u1_struct_0(A)) => ( ( r2_hidden(D,B) & r2_hidden(E,B) ) => r1_xreal_0(k2_metric_1(A,D,E),C) ) ) ) ) ) ) ) ), file(tbsp_1,d9_tbsp_1), [interesting(0.9),axiom,file(tbsp_1,d9_tbsp_1)]). fof(e3_46__jordan2c,plain,( ? [A] : ( m1_subset_1(A,k1_numbers) & ~ r1_xreal_0(A,0) & ! [B] : ( m1_subset_1(B,u1_struct_0(k14_euclid(c1_46__jordan2c))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k14_euclid(c1_46__jordan2c))) => ( ( r2_hidden(B,c2_46__jordan2c) & r2_hidden(C,c2_46__jordan2c) ) => r1_xreal_0(k4_metric_1(k14_euclid(c1_46__jordan2c),B,C),A) ) ) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_46__jordan2c,dt_c2_46__jordan2c,e2_46__jordan2c])],[cc1_arytm_3,cc1_finseq_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_k5_ordinal2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc14_finset_1,fc2_finseq_1,fc4_subset_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_finset_1,rc3_tbsp_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,free_g1_metric_1,reflexivity_r1_tarski,abstractness_v1_metric_1,existence_l1_struct_0,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_g1_metric_1,dt_k13_euclid,dt_k1_euclid,dt_k5_numbers,dt_l1_struct_0,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc2_xreal_0,cc3_xreal_0,cc4_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_euclid,fc1_struct_0,rc1_metric_1,rc1_subset_1,rc2_metric_1,rc2_subset_1,rc3_metric_1,rc3_struct_0,rc5_struct_0,t1_numerals,t1_real,t2_subset,t4_real,t5_subset,t6_boole,t8_boole,d1_euclid,commutativity_k4_metric_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_l1_metric_1,existence_m1_subset_1,redefinition_k4_metric_1,dt_k14_euclid,dt_k1_numbers,dt_k1_zfmisc_1,dt_k2_metric_1,dt_k4_metric_1,dt_l1_metric_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_46__jordan2c,dt_c2_46__jordan2c,cc6_membered,fc1_subset_1,fc2_membered,rc1_jordan2c,rc2_tbsp_1,rqLessOrEqual__r1_xreal_0__r0_r0,t1_subset,t3_subset,t4_subset,t7_boole,d7_euclid,spc0_numerals,spc0_boole,e2_46__jordan2c,d9_tbsp_1]), [interesting(0.8),file(jordan2c,e3_46__jordan2c),[file(jordan2c,e3_46__jordan2c)]]). fof(dt_c3_46__jordan2c,plain,( m1_subset_1(c3_46__jordan2c,k1_numbers) ), inference(consider,[status(thm),assumptions([dt_c1_46__jordan2c,dt_c2_46__jordan2c,e2_46__jordan2c])],[dh_c3_46__jordan2c,e3_46__jordan2c]), [interesting(0.8),file(jordan2c,c3_46__jordan2c),[file(jordan2c,c3_46__jordan2c)]]). fof(rqLessOrEqual__r1_xreal_0__r0_rm1,theorem,( ~ r1_xreal_0(0,k4_xcmplx_0(1)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_rm1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_rm1)]). fof(rqLessOrEqual__r1_xreal_0__r1_rm1,theorem,( ~ r1_xreal_0(1,k4_xcmplx_0(1)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_rm1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_rm1)]). fof(rqLessOrEqual__r1_xreal_0__rm1_r0,theorem,( r1_xreal_0(k4_xcmplx_0(1),0) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r0)]). fof(rqLessOrEqual__r1_xreal_0__rm1_r1,theorem,( r1_xreal_0(k4_xcmplx_0(1),1) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r1)]). fof(rqLessOrEqual__r1_xreal_0__rm1_rm1,theorem,( r1_xreal_0(k4_xcmplx_0(1),k4_xcmplx_0(1)) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm1_rm1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm1_rm1)]). fof(rqRealAdd__k2_xcmplx_0__r0_r0_r0,theorem,( k2_xcmplx_0(0,0) = 0 ), file(arithm,rqRealAdd__k2_xcmplx_0__r0_r0_r0), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r0_r0_r0)]). fof(rqRealAdd__k2_xcmplx_0__r0_r1_r1,theorem,( k2_xcmplx_0(0,1) = 1 ), file(arithm,rqRealAdd__k2_xcmplx_0__r0_r1_r1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r0_r1_r1)]). fof(rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,theorem,( k2_xcmplx_0(0,k4_xcmplx_0(1)) = k4_xcmplx_0(1) ), file(arithm,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1)]). fof(rqRealAdd__k2_xcmplx_0__r1_r0_r1,theorem,( k2_xcmplx_0(1,0) = 1 ), file(arithm,rqRealAdd__k2_xcmplx_0__r1_r0_r1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r1_r0_r1)]). fof(rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,theorem,( k2_xcmplx_0(k4_xcmplx_0(1),0) = k4_xcmplx_0(1) ), file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1)]). fof(rqRealNeg__k4_xcmplx_0__r0_r0,theorem,( k4_xcmplx_0(0) = 0 ), file(arithm,rqRealNeg__k4_xcmplx_0__r0_r0), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__r0_r0)]). fof(rqRealNeg__k4_xcmplx_0__rm1_r1,theorem,( k4_xcmplx_0(k4_xcmplx_0(1)) = 1 ), file(arithm,rqRealNeg__k4_xcmplx_0__rm1_r1), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__rm1_r1)]). fof(fc21_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & ~ v2_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc21_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc21_xreal_0)]). fof(fc22_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(B,A)) & v1_xreal_0(k3_xcmplx_0(B,A)) & ~ v2_xreal_0(k3_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc22_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc22_xreal_0)]). fof(fc24_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & ~ v3_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc24_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc24_xreal_0)]). fof(fc23_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & ~ v3_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc23_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc23_xreal_0)]). fof(spc2_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(A,k4_xcmplx_0(1)) = k4_xcmplx_0(A) ) ), file(arithm,spc2_arithm), [interesting(0.9),axiom,file(arithm,spc2_arithm)]). fof(spc5_arithm,theorem,( ! [A,B,C] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) & v1_xcmplx_0(C) ) => k3_xcmplx_0(k2_xcmplx_0(A,B),C) = k2_xcmplx_0(k3_xcmplx_0(A,C),k3_xcmplx_0(B,C)) ) ), file(arithm,spc5_arithm), [interesting(0.9),axiom,file(arithm,spc5_arithm)]). fof(spc7_arithm,theorem,( ! [A,B,C] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) & v1_xcmplx_0(C) ) => k3_xcmplx_0(k3_xcmplx_0(A,B),C) = k3_xcmplx_0(A,k3_xcmplx_0(B,C)) ) ), file(arithm,spc6_arithm), [interesting(0.9),axiom,file(arithm,spc6_arithm)]). fof(t2_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(A,0) = 0 ) ), file(arithm,t2_arithm), [interesting(0.9),axiom,file(arithm,t2_arithm)]). fof(t3_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(1,A) = A ) ), file(arithm,t3_arithm), [interesting(0.9),axiom,file(arithm,t3_arithm)]). fof(commutativity_k11_binop_2,theorem,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k11_binop_2(A,B) = k11_binop_2(B,A) ) ), file(binop_2,k11_binop_2), [interesting(0.9),axiom,file(binop_2,k11_binop_2)]). fof(commutativity_k3_xcmplx_0,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k3_xcmplx_0(A,B) = k3_xcmplx_0(B,A) ) ), file(xcmplx_0,k3_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k3_xcmplx_0)]). fof(redefinition_k11_binop_2,definition,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k11_binop_2(A,B) = k3_xcmplx_0(A,B) ) ), file(binop_2,k11_binop_2), [interesting(0.9),axiom,file(binop_2,k11_binop_2)]). fof(dt_k11_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k11_binop_2(A,B),k1_numbers) ) ), file(binop_2,k11_binop_2), [interesting(0.9),axiom,file(binop_2,k11_binop_2)]). fof(dt_k3_xcmplx_0,axiom,( $true ), file(xcmplx_0,k3_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k3_xcmplx_0)]). fof(fc4_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc4_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc4_xreal_0)]). fof(rqRealAdd__k2_xcmplx_0__rm1_r1_r0,theorem,( k2_xcmplx_0(k4_xcmplx_0(1),1) = 0 ), file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r1_r0), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r1_r0)]). fof(rqRealMult__k3_xcmplx_0__r0_r0_r0,theorem,( k3_xcmplx_0(0,0) = 0 ), file(arithm,rqRealMult__k3_xcmplx_0__r0_r0_r0), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r0_r0_r0)]). fof(rqRealMult__k3_xcmplx_0__r0_r1_r0,theorem,( k3_xcmplx_0(0,1) = 0 ), file(arithm,rqRealMult__k3_xcmplx_0__r0_r1_r0), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r0_r1_r0)]). fof(rqRealMult__k3_xcmplx_0__r1_r0_r0,theorem,( k3_xcmplx_0(1,0) = 0 ), file(arithm,rqRealMult__k3_xcmplx_0__r1_r0_r0), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r1_r0_r0)]). fof(dt_k8_square_1,axiom,( ! [A] : ( v1_xreal_0(A) => v1_xreal_0(k8_square_1(A)) ) ), file(square_1,k8_square_1), [interesting(0.9),axiom,file(square_1,k8_square_1)]). fof(redefinition_k9_square_1,definition,( ! [A] : ( m1_subset_1(A,k1_numbers) => k9_square_1(A) = k8_square_1(A) ) ), file(square_1,k9_square_1), [interesting(0.9),axiom,file(square_1,k9_square_1)]). fof(dt_k9_square_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => m1_subset_1(k9_square_1(A),k1_numbers) ) ), file(square_1,k9_square_1), [interesting(0.9),axiom,file(square_1,k9_square_1)]). fof(t94_square_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(0,A) & r1_xreal_0(A,B) ) => r1_xreal_0(k8_square_1(A),k8_square_1(B)) ) ) ) ), file(square_1,t94_square_1), [interesting(0.9),axiom,file(square_1,t94_square_1)]). fof(e17_46__jordan2c,plain,( r1_xreal_0(k9_square_1(1),k9_square_1(c1_46__jordan2c)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_46__jordan2c,dt_c2_46__jordan2c,e1_46__jordan2c])],[existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_arytm_3,cc1_finseq_1,cc1_relset_1,cc2_arytm_3,cc2_funct_1,fc14_finset_1,fc4_subset_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,existence_l1_struct_0,dt_g1_metric_1,dt_k13_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_l1_metric_1,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_struct_0,fc2_finseq_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_metric_1,rc2_metric_1,rc2_xreal_0,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc5_struct_0,t1_subset,t2_real,t3_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_finseq_2,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k1_numbers,dt_k1_zfmisc_1,dt_k4_finseq_2,dt_k5_numbers,dt_m1_finseq_2,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_finset_1,cc1_funct_1,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_euclid,fc1_subset_1,fc2_membered,rc1_subset_1,rc1_xreal_0,rc2_subset_1,t1_numerals,t1_real,t2_subset,t3_subset,t4_real,t6_boole,t7_boole,t8_boole,d7_euclid,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k9_square_1,dt_k1_euclid,dt_k8_square_1,dt_k9_square_1,dt_c1_46__jordan2c,dt_c2_46__jordan2c,cc2_xreal_0,rqLessOrEqual__r1_xreal_0__r1_r0,d1_euclid,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e1_46__jordan2c,t94_square_1,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.8),file(jordan2c,e17_46__jordan2c),[file(jordan2c,e17_46__jordan2c)]]). fof(t31_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,0) & r1_xreal_0(k2_xcmplx_0(B,A),B) ) ) ) ), file(xreal_1,t31_xreal_1), [interesting(0.9),axiom,file(xreal_1,t31_xreal_1)]). fof(e14_46__jordan2c,plain,( ~ r1_xreal_0(k9_binop_2(c3_46__jordan2c,1),c3_46__jordan2c) ), inference(mizar_by,[status(thm),assumptions([dt_c1_46__jordan2c,dt_c2_46__jordan2c,e2_46__jordan2c])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_finseq_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_subset_1,fc2_finseq_1,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_finset_1,rc1_membered,rc1_subset_1,rc2_subset_1,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc2_membered,fc8_xreal_0,rc1_xreal_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,spc6_arithm,t1_arithm,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k2_xcmplx_0,commutativity_k9_binop_2,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k9_binop_2,dt_k2_xcmplx_0,dt_k9_binop_2,dt_c3_46__jordan2c,cc2_xreal_0,fc3_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r1,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,t31_xreal_1,rqLessOrEqual__r1_xreal_0__r1_r0]), [interesting(0.8),file(jordan2c,e14_46__jordan2c),[file(jordan2c,e14_46__jordan2c)]]). fof(e4_46__jordan2c,plain, ( ~ r1_xreal_0(c3_46__jordan2c,0) & ! [A] : ( m1_subset_1(A,u1_struct_0(k14_euclid(c1_46__jordan2c))) => ! [B] : ( m1_subset_1(B,u1_struct_0(k14_euclid(c1_46__jordan2c))) => ( ( r2_hidden(A,c2_46__jordan2c) & r2_hidden(B,c2_46__jordan2c) ) => r1_xreal_0(k4_metric_1(k14_euclid(c1_46__jordan2c),A,B),c3_46__jordan2c) ) ) ) ), inference(consider,[status(thm),assumptions([dt_c1_46__jordan2c,dt_c2_46__jordan2c,e2_46__jordan2c])],[dh_c3_46__jordan2c,e3_46__jordan2c]), [interesting(0.8),file(jordan2c,e4_46__jordan2c),[file(jordan2c,e4_46__jordan2c)]]). fof(e15_46__jordan2c,plain,( ~ r1_xreal_0(k9_binop_2(c3_46__jordan2c,1),0) ), inference(mizar_by,[status(thm),assumptions([dt_c1_46__jordan2c,dt_c2_46__jordan2c,e2_46__jordan2c])],[cc1_arytm_3,cc1_finseq_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_k5_ordinal2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_finset_1,fc1_struct_0,fc2_finseq_1,fc4_subset_1,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_finset_1,rc1_membered,rc1_xreal_0,rc2_metric_1,rc2_xreal_0,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc5_struct_0,spc6_arithm,t1_arithm,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,free_g1_metric_1,commutativity_k2_xcmplx_0,abstractness_v1_metric_1,existence_l1_metric_1,existence_l1_struct_0,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_g1_metric_1,dt_k13_euclid,dt_k1_euclid,dt_k1_numbers,dt_k1_zfmisc_1,dt_k2_metric_1,dt_k2_xcmplx_0,dt_k5_numbers,dt_l1_metric_1,dt_l1_struct_0,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_xreal_0,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_euclid,fc1_subset_1,fc2_membered,fc3_xreal_0,fc8_xreal_0,rc1_metric_1,rc1_subset_1,rc2_subset_1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,t1_numerals,t1_real,t2_subset,t3_subset,t4_real,t4_subset,t5_subset,t6_boole,t8_boole,d1_euclid,commutativity_k4_metric_1,commutativity_k9_binop_2,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,redefinition_k4_metric_1,redefinition_k9_binop_2,dt_k14_euclid,dt_k4_metric_1,dt_k9_binop_2,dt_m1_subset_1,dt_u1_struct_0,dt_c1_46__jordan2c,dt_c2_46__jordan2c,dt_c3_46__jordan2c,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,t1_subset,t7_boole,d7_euclid,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e14_46__jordan2c,e4_46__jordan2c]), [interesting(0.8),file(jordan2c,e15_46__jordan2c),[file(jordan2c,e15_46__jordan2c)]]). fof(t66_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ( r1_xreal_0(A,B) & r1_xreal_0(0,C) ) => r1_xreal_0(k3_xcmplx_0(A,C),k3_xcmplx_0(B,C)) ) ) ) ) ), file(xreal_1,t66_xreal_1), [interesting(0.9),axiom,file(xreal_1,t66_xreal_1)]). fof(t83_square_1,theorem,( k9_square_1(1) = 1 ), file(square_1,t83_square_1), [interesting(0.9),axiom,file(square_1,t83_square_1)]). fof(rqRealMult__k3_xcmplx_0__r1_r1_r1,theorem,( k3_xcmplx_0(1,1) = 1 ), file(arithm,rqRealMult__k3_xcmplx_0__r1_r1_r1), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r1_r1_r1)]). fof(e18_46__jordan2c,plain,( r1_xreal_0(k11_binop_2(k9_binop_2(c3_46__jordan2c,1),1),k11_binop_2(k9_binop_2(c3_46__jordan2c,1),k9_square_1(c1_46__jordan2c))) ), inference(mizar_by,[status(thm),assumptions([e1_46__jordan2c,dt_c1_46__jordan2c,dt_c2_46__jordan2c,e2_46__jordan2c])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_finseq_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_subset_1,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_finseq_1,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_finset_1,rc1_membered,rc1_subset_1,rc2_subset_1,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_xcmplx_0,dt_k5_numbers,dt_k8_square_1,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc23_xreal_0,fc2_membered,fc3_xreal_0,fc8_xreal_0,rc1_xreal_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,spc5_arithm,spc6_arithm,spc7_arithm,t1_arithm,t1_numerals,t1_real,t2_arithm,t2_subset,t3_arithm,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k11_binop_2,commutativity_k3_xcmplx_0,commutativity_k9_binop_2,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k11_binop_2,redefinition_k9_binop_2,redefinition_k9_square_1,dt_k11_binop_2,dt_k3_xcmplx_0,dt_k9_binop_2,dt_k9_square_1,dt_c1_46__jordan2c,dt_c3_46__jordan2c,cc2_xreal_0,fc4_xreal_0,rqLessOrEqual__r1_xreal_0__r1_r0,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e17_46__jordan2c,e15_46__jordan2c,t66_xreal_1,t83_square_1,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r0]), [interesting(0.8),file(jordan2c,e18_46__jordan2c),[file(jordan2c,e18_46__jordan2c)]]). fof(rqRealAdd__k2_xcmplx_0__r1_rm1_r0,theorem,( k2_xcmplx_0(1,k4_xcmplx_0(1)) = 0 ), file(arithm,rqRealAdd__k2_xcmplx_0__r1_rm1_r0), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r1_rm1_r0)]). fof(projectivity_k16_complex1,theorem,( ! [A] : ( v1_xcmplx_0(A) => k16_complex1(k16_complex1(A)) = k16_complex1(A) ) ), file(complex1,k16_complex1), [interesting(0.9),axiom,file(complex1,k16_complex1)]). fof(projectivity_k18_complex1,theorem,( ! [A] : ( v1_xcmplx_0(A) => k18_complex1(k18_complex1(A)) = k18_complex1(A) ) ), file(complex1,k18_complex1), [interesting(0.9),axiom,file(complex1,k18_complex1)]). fof(redefinition_k18_complex1,definition,( ! [A] : ( v1_xcmplx_0(A) => k18_complex1(A) = k16_complex1(A) ) ), file(complex1,k18_complex1), [interesting(0.9),axiom,file(complex1,k18_complex1)]). fof(dt_k16_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_xcmplx_0(k16_complex1(A)) ) ), file(complex1,k16_complex1), [interesting(0.9),axiom,file(complex1,k16_complex1)]). fof(dt_k18_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => m1_subset_1(k18_complex1(A),k1_numbers) ) ), file(complex1,k18_complex1), [interesting(0.9),axiom,file(complex1,k18_complex1)]). fof(rqRealNeg__k4_xcmplx_0__r1_rm1,theorem,( k4_xcmplx_0(1) = k4_xcmplx_0(1) ), file(arithm,rqRealNeg__k4_xcmplx_0__r1_rm1), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__r1_rm1)]). fof(existence_m1_finseq_1,axiom,( ! [A] : ? [B] : m1_finseq_1(B,A) ), file(finseq_1,m1_finseq_1), [interesting(0.9),axiom,file(finseq_1,m1_finseq_1)]). fof(dt_m1_finseq_1,axiom,( ! [A,B] : ( m1_finseq_1(B,A) => ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) ) ), file(finseq_1,m1_finseq_1), [interesting(0.9),axiom,file(finseq_1,m1_finseq_1)]). fof(rc2_goboard1,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_finseq_1(B,A) & ~ v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v1_finset_1(B) & v1_finseq_1(B) ) ) ), file(goboard1,rc2_goboard1), [interesting(0.9),axiom,file(goboard1,rc2_goboard1)]). fof(rc4_finseq_1,theorem,( ! [A] : ? [B] : ( m1_finseq_1(B,A) & v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v2_funct_1(B) & v1_finset_1(B) & v1_finseq_1(B) ) ), file(finseq_1,rc4_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc4_finseq_1)]). fof(free_g1_pre_topc,definition,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) => ! [C,D] : ( g1_pre_topc(A,B) = g1_pre_topc(C,D) => ( A = C & B = D ) ) ) ), file(pre_topc,g1_pre_topc), [interesting(0.9),axiom,file(pre_topc,g1_pre_topc)]). fof(existence_m2_finseq_1,axiom,( ! [A] : ? [B] : m2_finseq_1(B,A) ), file(finseq_1,m2_finseq_1), [interesting(0.9),axiom,file(finseq_1,m2_finseq_1)]). fof(redefinition_m2_finseq_1,definition,( ! [A,B] : ( m2_finseq_1(B,A) <=> m1_finseq_1(B,A) ) ), file(finseq_1,m2_finseq_1), [interesting(0.9),axiom,file(finseq_1,m2_finseq_1)]). fof(dt_g1_pre_topc,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) => ( v1_pre_topc(g1_pre_topc(A,B)) & l1_pre_topc(g1_pre_topc(A,B)) ) ) ), file(pre_topc,g1_pre_topc), [interesting(0.9),axiom,file(pre_topc,g1_pre_topc)]). fof(dt_k2_finseq_2,axiom,( ! [A,B] : ( v4_ordinal2(A) => ( v1_relat_1(k2_finseq_2(A,B)) & v1_funct_1(k2_finseq_2(A,B)) & v1_finseq_1(k2_finseq_2(A,B)) ) ) ), file(finseq_2,k2_finseq_2), [interesting(0.9),axiom,file(finseq_2,k2_finseq_2)]). fof(dt_m2_finseq_1,axiom,( ! [A,B] : ( m2_finseq_1(B,A) => ( v1_funct_1(B) & v1_finseq_1(B) & m2_relset_1(B,k5_numbers,A) ) ) ), file(finseq_1,m2_finseq_1), [interesting(0.9),axiom,file(finseq_1,m2_finseq_1)]). fof(dt_u1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => m1_subset_1(u1_pre_topc(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ), file(pre_topc,u1_pre_topc), [interesting(0.9),axiom,file(pre_topc,u1_pre_topc)]). fof(fc3_pcomps_1,theorem,( ! [A] : ( l1_metric_1(A) => ( v1_pre_topc(k5_pcomps_1(A)) & v2_pre_topc(k5_pcomps_1(A)) ) ) ), file(pcomps_1,fc3_pcomps_1), [interesting(0.9),axiom,file(pcomps_1,fc3_pcomps_1)]). fof(fc4_pcomps_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l1_metric_1(A) ) => ( ~ v3_struct_0(k5_pcomps_1(A)) & v1_pre_topc(k5_pcomps_1(A)) & v2_pre_topc(k5_pcomps_1(A)) ) ) ), file(pcomps_1,fc4_pcomps_1), [interesting(0.9),axiom,file(pcomps_1,fc4_pcomps_1)]). fof(abstractness_v1_pre_topc,theorem,( ! [A] : ( l1_pre_topc(A) => ( v1_pre_topc(A) => A = g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) ) ) ), file(pre_topc,v1_pre_topc), [interesting(0.9),axiom,file(pre_topc,v1_pre_topc)]). fof(existence_l1_pre_topc,axiom,( ? [A] : l1_pre_topc(A) ), file(pre_topc,l1_pre_topc), [interesting(0.9),axiom,file(pre_topc,l1_pre_topc)]). fof(existence_m2_finseq_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & m1_finseq_2(B,A) ) => ? [C] : m2_finseq_2(C,A,B) ) ), file(finseq_2,m2_finseq_2), [interesting(0.9),axiom,file(finseq_2,m2_finseq_2)]). fof(redefinition_m2_finseq_2,definition,( ! [A,B] : ( ( ~ v1_xboole_0(B) & m1_finseq_2(B,A) ) => ! [C] : ( m2_finseq_2(C,A,B) <=> m1_subset_1(C,B) ) ) ), file(finseq_2,m2_finseq_2), [interesting(0.9),axiom,file(finseq_2,m2_finseq_2)]). fof(dt_k3_jordan2c,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => m2_finseq_1(k3_jordan2c(A),k1_numbers) ) ), file(jordan2c,k3_jordan2c), [interesting(0.9),axiom,file(jordan2c,k3_jordan2c)]). fof(dt_k5_pcomps_1,axiom,( ! [A] : ( l1_metric_1(A) => l1_pre_topc(k5_pcomps_1(A)) ) ), file(pcomps_1,k5_pcomps_1), [interesting(0.9),axiom,file(pcomps_1,k5_pcomps_1)]). fof(dt_l1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => l1_struct_0(A) ) ), file(pre_topc,l1_pre_topc), [interesting(0.9),axiom,file(pre_topc,l1_pre_topc)]). fof(dt_m2_finseq_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & m1_finseq_2(B,A) ) => ! [C] : ( m2_finseq_2(C,A,B) => m2_finseq_1(C,A) ) ) ), file(finseq_2,m2_finseq_2), [interesting(0.9),axiom,file(finseq_2,m2_finseq_2)]). fof(rc1_pre_topc,theorem,( ? [A] : ( l1_pre_topc(A) & v1_pre_topc(A) ) ), file(pre_topc,rc1_pre_topc), [interesting(0.9),axiom,file(pre_topc,rc1_pre_topc)]). fof(rc2_pcomps_1,theorem,( ? [A] : ( l1_pre_topc(A) & ~ v3_struct_0(A) & v1_pre_topc(A) & v2_pre_topc(A) & v3_compts_1(A) ) ), file(pcomps_1,rc2_pcomps_1), [interesting(0.9),axiom,file(pcomps_1,rc2_pcomps_1)]). fof(rc2_pre_topc,theorem,( ? [A] : ( l1_pre_topc(A) & ~ v3_struct_0(A) & v1_pre_topc(A) & v2_pre_topc(A) ) ), file(pre_topc,rc2_pre_topc), [interesting(0.9),axiom,file(pre_topc,rc2_pre_topc)]). fof(d7_jordan2c,definition,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k3_jordan2c(A) = k2_finseq_2(A,1) ) ), file(jordan2c,d7_jordan2c), [interesting(0.9),axiom,file(jordan2c,d7_jordan2c)]). fof(redefinition_k4_jordan2c,definition,( ! [A] : ( m1_subset_1(A,k5_numbers) => k4_jordan2c(A) = k3_jordan2c(A) ) ), file(jordan2c,k4_jordan2c), [interesting(0.9),axiom,file(jordan2c,k4_jordan2c)]). fof(dt_k15_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_pre_topc(k15_euclid(A)) & v2_pre_topc(k15_euclid(A)) & l1_pre_topc(k15_euclid(A)) ) ) ), file(euclid,k15_euclid), [interesting(0.9),axiom,file(euclid,k15_euclid)]). fof(dt_k4_jordan2c,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => m2_finseq_2(k4_jordan2c(A),k1_numbers,k1_euclid(A)) ) ), file(jordan2c,k4_jordan2c), [interesting(0.9),axiom,file(jordan2c,k4_jordan2c)]). fof(fc2_euclid,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( ~ v3_struct_0(k15_euclid(A)) & v1_pre_topc(k15_euclid(A)) & v2_pre_topc(k15_euclid(A)) ) ) ), file(euclid,fc2_euclid), [interesting(0.9),axiom,file(euclid,fc2_euclid)]). fof(fc2_topreal1,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( ~ v3_struct_0(k15_euclid(A)) & v1_pre_topc(k15_euclid(A)) & v2_pre_topc(k15_euclid(A)) & v3_compts_1(k15_euclid(A)) ) ) ), file(topreal1,fc2_topreal1), [interesting(0.9),axiom,file(topreal1,fc2_topreal1)]). fof(d8_euclid,definition,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k15_euclid(A) = k5_pcomps_1(k14_euclid(A)) ) ), file(euclid,d8_euclid), [interesting(0.9),axiom,file(euclid,d8_euclid)]). fof(dt_k5_jordan2c,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => m1_subset_1(k5_jordan2c(A),u1_struct_0(k15_euclid(A))) ) ), file(jordan2c,k5_jordan2c), [interesting(0.9),axiom,file(jordan2c,k5_jordan2c)]). fof(dt_k5_toprns_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) ) => m1_subset_1(k5_toprns_1(A,B),k1_numbers) ) ), file(toprns_1,k5_toprns_1), [interesting(0.9),axiom,file(toprns_1,k5_toprns_1)]). fof(d8_jordan2c,definition,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k5_jordan2c(A) = k4_jordan2c(A) ) ), file(jordan2c,d8_jordan2c), [interesting(0.9),axiom,file(jordan2c,d8_jordan2c)]). fof(commutativity_k4_real_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k4_real_1(A,B) = k4_real_1(B,A) ) ), file(real_1,k4_real_1), [interesting(0.9),axiom,file(real_1,k4_real_1)]). fof(redefinition_k4_real_1,definition,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k4_real_1(A,B) = k3_xcmplx_0(A,B) ) ), file(real_1,k4_real_1), [interesting(0.9),axiom,file(real_1,k4_real_1)]). fof(dt_k18_euclid,axiom,( ! [A,B,C] : ( ( v1_xreal_0(A) & m1_subset_1(B,k5_numbers) & m1_subset_1(C,u1_struct_0(k15_euclid(B))) ) => m1_subset_1(k18_euclid(A,B,C),u1_struct_0(k15_euclid(B))) ) ), file(euclid,k18_euclid), [interesting(0.9),axiom,file(euclid,k18_euclid)]). fof(dt_k4_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k4_real_1(A,B),k1_numbers) ) ), file(real_1,k4_real_1), [interesting(0.9),axiom,file(real_1,k4_real_1)]). fof(redefinition_k4_finseqop,definition,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,k5_numbers) & m1_subset_1(C,A) ) => k4_finseqop(A,B,C) = k2_finseq_2(B,C) ) ), file(finseqop,k4_finseqop), [interesting(0.9),axiom,file(finseqop,k4_finseqop)]). fof(dt_k4_finseqop,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,k5_numbers) & m1_subset_1(C,A) ) => m2_finseq_2(k4_finseqop(A,B,C),A,k4_finseq_2(B,A)) ) ), file(finseqop,k4_finseqop), [interesting(0.9),axiom,file(finseqop,k4_finseqop)]). fof(dt_k4_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => m2_finseq_1(k4_euclid(A),k1_numbers) ) ), file(euclid,k4_euclid), [interesting(0.9),axiom,file(euclid,k4_euclid)]). fof(d4_euclid,definition,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k4_euclid(A) = k4_finseqop(k1_numbers,A,0) ) ), file(euclid,d4_euclid), [interesting(0.9),axiom,file(euclid,d4_euclid)]). fof(redefinition_k5_euclid,definition,( ! [A] : ( m1_subset_1(A,k5_numbers) => k5_euclid(A) = k4_euclid(A) ) ), file(euclid,k5_euclid), [interesting(0.9),axiom,file(euclid,k5_euclid)]). fof(dt_k5_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => m2_finseq_2(k5_euclid(A),k1_numbers,k1_euclid(A)) ) ), file(euclid,k5_euclid), [interesting(0.9),axiom,file(euclid,k5_euclid)]). fof(dt_k16_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => m1_subset_1(k16_euclid(A),u1_struct_0(k15_euclid(A))) ) ), file(euclid,k16_euclid), [interesting(0.9),axiom,file(euclid,k16_euclid)]). fof(dt_k19_euclid,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) ) => m1_subset_1(k19_euclid(A,B),u1_struct_0(k15_euclid(A))) ) ), file(euclid,k19_euclid), [interesting(0.9),axiom,file(euclid,k19_euclid)]). fof(dt_k20_euclid,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & m1_subset_1(C,u1_struct_0(k15_euclid(A))) ) => m1_subset_1(k20_euclid(A,B,C),u1_struct_0(k15_euclid(A))) ) ), file(euclid,k20_euclid), [interesting(0.9),axiom,file(euclid,k20_euclid)]). fof(d9_euclid,definition,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k16_euclid(A) = k5_euclid(A) ) ), file(euclid,d9_euclid), [interesting(0.9),axiom,file(euclid,d9_euclid)]). fof(de_c4_46__jordan2c,definition,( c4_46__jordan2c = k18_euclid(k9_binop_2(c3_46__jordan2c,1),c1_46__jordan2c,k5_jordan2c(c1_46__jordan2c)) ), introduced(definition,[new_symbol(c4_46__jordan2c),file(jordan2c,c4_46__jordan2c)]), [interesting(0.8),axiom,file(jordan2c,c4_46__jordan2c)]). fof(t13_topreal3,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => u1_struct_0(k15_euclid(A)) = u1_struct_0(k14_euclid(A)) ) ), file(topreal3,t13_topreal3), [interesting(0.9),axiom,file(topreal3,t13_topreal3)]). fof(e5_46__jordan2c,plain,( m1_subset_1(k18_euclid(k9_binop_2(c3_46__jordan2c,1),c1_46__jordan2c,k5_jordan2c(c1_46__jordan2c)),u1_struct_0(k14_euclid(c1_46__jordan2c))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_46__jordan2c,dt_c2_46__jordan2c,e2_46__jordan2c])],[existence_m1_finseq_1,dt_m1_finseq_1,rc2_goboard1,rc4_finseq_1,existence_m2_finseq_1,redefinition_m2_finseq_1,dt_k2_finseq_2,dt_m2_finseq_1,cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_pre_topc,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_finseq_2,existence_m2_relset_1,redefinition_m2_finseq_2,redefinition_m2_relset_1,dt_g1_pre_topc,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k3_jordan2c,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_finseq_2,dt_m2_relset_1,dt_u1_metric_1,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_finset_1,fc2_finseq_1,fc4_subset_1,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_arytm_3,rc1_finset_1,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,spc6_arithm,t1_subset,t4_subset,t5_subset,d7_jordan2c,free_g1_metric_1,commutativity_k2_xcmplx_0,abstractness_v1_metric_1,abstractness_v1_pre_topc,existence_l1_metric_1,existence_l1_pre_topc,existence_l1_struct_0,redefinition_k4_jordan2c,dt_g1_metric_1,dt_k13_euclid,dt_k1_euclid,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k4_jordan2c,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_metric_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_struct_0,fc1_subset_1,fc3_pcomps_1,fc3_xreal_0,fc4_pcomps_1,fc5_membered,fc8_xreal_0,rc1_metric_1,rc1_pre_topc,rc1_subset_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc3_metric_1,rc3_struct_0,rc5_struct_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,d1_euclid,commutativity_k9_binop_2,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_k9_binop_2,redefinition_m2_subset_1,dt_k14_euclid,dt_k15_euclid,dt_k18_euclid,dt_k1_numbers,dt_k5_jordan2c,dt_k5_numbers,dt_k9_binop_2,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_46__jordan2c,dt_c3_46__jordan2c,fc1_euclid,fc2_euclid,fc2_membered,fc2_topreal1,d7_euclid,d8_euclid,d8_jordan2c,spc1_numerals,spc1_boole,t13_topreal3]), [interesting(0.8),file(jordan2c,e5_46__jordan2c),[file(jordan2c,e5_46__jordan2c)]]). fof(dt_c4_46__jordan2c,plain,( m1_subset_1(c4_46__jordan2c,u1_struct_0(k14_euclid(c1_46__jordan2c))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_46__jordan2c,dt_c2_46__jordan2c,e2_46__jordan2c])],[existence_m1_finseq_1,dt_m1_finseq_1,rc2_goboard1,rc4_finseq_1,free_g1_pre_topc,reflexivity_r1_tarski,existence_m2_finseq_1,redefinition_m2_finseq_1,dt_g1_pre_topc,dt_k2_finseq_2,dt_m2_finseq_1,dt_u1_pre_topc,cc1_arytm_3,cc1_finseq_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_finseq_2,existence_m2_relset_1,redefinition_m2_finseq_2,redefinition_m2_relset_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k3_jordan2c,dt_k4_finseq_2,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_finseq_2,dt_m2_relset_1,dt_u1_metric_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_finset_1,fc1_struct_0,fc1_subset_1,fc2_finseq_1,fc3_pcomps_1,fc4_pcomps_1,fc4_subset_1,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_finset_1,rc1_membered,rc1_pre_topc,rc1_subset_1,rc1_xreal_0,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc5_struct_0,spc6_arithm,t1_subset,t3_subset,t4_subset,t5_subset,d7_jordan2c,free_g1_metric_1,commutativity_k2_xcmplx_0,abstractness_v1_metric_1,existence_l1_metric_1,existence_l1_struct_0,existence_m2_subset_1,redefinition_k4_jordan2c,redefinition_k5_numbers,redefinition_m2_subset_1,dt_g1_metric_1,dt_k13_euclid,dt_k15_euclid,dt_k1_euclid,dt_k1_numbers,dt_k2_xcmplx_0,dt_k4_jordan2c,dt_k5_numbers,dt_l1_metric_1,dt_l1_struct_0,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_euclid,fc2_euclid,fc2_membered,fc2_topreal1,fc3_xreal_0,fc8_xreal_0,rc1_metric_1,t2_subset,t6_boole,t7_boole,t8_boole,d8_euclid,d1_euclid,commutativity_k9_binop_2,existence_m1_subset_1,redefinition_k9_binop_2,dt_k14_euclid,dt_k18_euclid,dt_k5_jordan2c,dt_k9_binop_2,dt_m1_subset_1,dt_u1_struct_0,dt_c1_46__jordan2c,dt_c3_46__jordan2c,d7_euclid,d8_jordan2c,spc1_numerals,spc1_boole,de_c4_46__jordan2c,e5_46__jordan2c]), [interesting(0.8),file(jordan2c,c4_46__jordan2c),[file(jordan2c,c4_46__jordan2c)]]). fof(de_c5_46__jordan2c,definition,( c5_46__jordan2c = k16_euclid(c1_46__jordan2c) ), introduced(definition,[new_symbol(c5_46__jordan2c),file(jordan2c,c5_46__jordan2c)]), [interesting(0.8),axiom,file(jordan2c,c5_46__jordan2c)]). fof(e7_46__jordan2c,plain,( m1_subset_1(k16_euclid(c1_46__jordan2c),u1_struct_0(k14_euclid(c1_46__jordan2c))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_46__jordan2c])],[existence_m1_finseq_1,dt_k2_finseq_2,dt_m1_finseq_1,cc1_finseq_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_finseq_1,rc1_jordan2c,rc2_finseq_1,rc2_finset_1,rc2_goboard1,rc2_tbsp_1,rc2_xreal_0,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc3_xreal_0,rc4_finseq_1,rc4_funct_1,rc4_xreal_0,rc6_finseq_1,rc8_finseq_1,free_g1_pre_topc,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_m2_finseq_1,redefinition_k4_finseqop,redefinition_m2_finseq_1,dt_g1_pre_topc,dt_k1_xboole_0,dt_k4_finseqop,dt_m2_finseq_1,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_finset_1,cc2_funct_1,cc2_membered,cc2_tbsp_1,cc2_xreal_0,cc3_membered,cc7_xreal_0,fc14_finset_1,fc2_finseq_1,fc6_membered,rc1_arytm_3,rc1_finset_1,rc1_funct_1,rc1_membered,rc1_xreal_0,rc2_funct_1,rc3_finset_1,rc4_finset_1,rc7_finseq_1,spc0_boole,t1_numerals,t1_subset,t4_subset,t5_subset,spc0_numerals,spc0_boole,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_finseq_2,existence_m2_relset_1,redefinition_m2_finseq_2,redefinition_m2_relset_1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k4_euclid,dt_k4_finseq_2,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_finseq_2,dt_m2_relset_1,dt_u1_metric_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc1_relset_1,cc3_arytm_3,cc4_membered,cc6_membered,cc9_membered,fc1_struct_0,fc1_subset_1,fc3_pcomps_1,fc4_pcomps_1,fc4_subset_1,fc5_membered,rc1_pre_topc,rc1_subset_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc3_metric_1,rc3_struct_0,rc5_struct_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,d4_euclid,free_g1_metric_1,abstractness_v1_metric_1,existence_l1_metric_1,existence_l1_struct_0,existence_m2_subset_1,redefinition_k5_euclid,redefinition_k5_numbers,redefinition_m2_subset_1,dt_g1_metric_1,dt_k13_euclid,dt_k15_euclid,dt_k1_euclid,dt_k1_numbers,dt_k5_euclid,dt_k5_numbers,dt_l1_metric_1,dt_l1_struct_0,dt_m2_subset_1,fc1_euclid,fc2_euclid,fc2_membered,fc2_topreal1,rc1_metric_1,d8_euclid,d1_euclid,existence_m1_subset_1,dt_k14_euclid,dt_k16_euclid,dt_m1_subset_1,dt_u1_struct_0,dt_c1_46__jordan2c,d7_euclid,d9_euclid]), [interesting(0.8),file(jordan2c,e7_46__jordan2c),[file(jordan2c,e7_46__jordan2c)]]). fof(dt_c5_46__jordan2c,plain,( m1_subset_1(c5_46__jordan2c,u1_struct_0(k14_euclid(c1_46__jordan2c))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_46__jordan2c])],[existence_m1_finseq_1,dt_k2_finseq_2,dt_m1_finseq_1,cc1_finseq_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_finseq_1,rc1_jordan2c,rc2_finseq_1,rc2_finset_1,rc2_goboard1,rc2_tbsp_1,rc2_xreal_0,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc3_xreal_0,rc4_finseq_1,rc4_funct_1,rc4_xreal_0,rc6_finseq_1,rc8_finseq_1,free_g1_pre_topc,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_m2_finseq_1,redefinition_k4_finseqop,redefinition_m2_finseq_1,dt_g1_pre_topc,dt_k1_xboole_0,dt_k4_finseqop,dt_m2_finseq_1,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_finset_1,cc2_funct_1,cc2_membered,cc2_tbsp_1,cc2_xreal_0,cc3_membered,cc7_xreal_0,fc14_finset_1,fc2_finseq_1,fc6_membered,rc1_arytm_3,rc1_finset_1,rc1_funct_1,rc1_membered,rc1_xreal_0,rc2_funct_1,rc3_finset_1,rc4_finset_1,rc7_finseq_1,spc0_boole,t1_numerals,t1_subset,t4_subset,t5_subset,spc0_numerals,spc0_boole,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_finseq_2,existence_m2_relset_1,redefinition_m2_finseq_2,redefinition_m2_relset_1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k4_euclid,dt_k4_finseq_2,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_finseq_2,dt_m2_relset_1,dt_u1_metric_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc1_relset_1,cc3_arytm_3,cc4_membered,cc6_membered,cc9_membered,fc1_struct_0,fc1_subset_1,fc3_pcomps_1,fc4_pcomps_1,fc4_subset_1,fc5_membered,rc1_pre_topc,rc1_subset_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc3_metric_1,rc3_struct_0,rc5_struct_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,d4_euclid,free_g1_metric_1,abstractness_v1_metric_1,existence_l1_metric_1,existence_l1_struct_0,existence_m2_subset_1,redefinition_k5_euclid,redefinition_k5_numbers,redefinition_m2_subset_1,dt_g1_metric_1,dt_k13_euclid,dt_k15_euclid,dt_k1_euclid,dt_k1_numbers,dt_k5_euclid,dt_k5_numbers,dt_l1_metric_1,dt_l1_struct_0,dt_m2_subset_1,fc1_euclid,fc2_euclid,fc2_membered,fc2_topreal1,rc1_metric_1,d8_euclid,d1_euclid,existence_m1_subset_1,dt_k14_euclid,dt_k16_euclid,dt_m1_subset_1,dt_u1_struct_0,dt_c1_46__jordan2c,d7_euclid,d9_euclid,de_c5_46__jordan2c,e7_46__jordan2c]), [interesting(0.8),file(jordan2c,c5_46__jordan2c),[file(jordan2c,c5_46__jordan2c)]]). fof(e8_46__jordan2c,plain,( r2_hidden(c5_46__jordan2c,c2_46__jordan2c) ), inference(mizar_by,[status(thm),assumptions([dt_c2_46__jordan2c,dt_c1_46__jordan2c,e1_46__jordan2c])],[existence_m1_finseq_1,dt_k2_finseq_2,dt_m1_finseq_1,rc2_goboard1,rc4_finseq_1,free_g1_pre_topc,existence_m2_finseq_1,redefinition_k4_finseqop,redefinition_m2_finseq_1,dt_g1_pre_topc,dt_k4_finseqop,dt_m2_finseq_1,dt_u1_pre_topc,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r0,spc0_boole,spc0_numerals,t1_numerals,spc0_numerals,spc0_boole,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_m1_relset_1,existence_m2_finseq_2,existence_m2_relset_1,redefinition_m2_finseq_2,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_euclid,dt_k5_pcomps_1,dt_l1_pre_topc,dt_m1_relset_1,dt_m2_finseq_2,dt_m2_relset_1,dt_u1_metric_1,cc1_arytm_3,cc1_finseq_1,cc1_relset_1,cc2_arytm_3,cc2_funct_1,fc14_finset_1,fc3_pcomps_1,fc4_pcomps_1,fc4_subset_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc1_pre_topc,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_pcomps_1,rc2_pre_topc,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,d4_euclid,free_g1_metric_1,reflexivity_r1_tarski,abstractness_v1_metric_1,existence_l1_metric_1,existence_l1_struct_0,redefinition_k5_euclid,dt_g1_metric_1,dt_k13_euclid,dt_k15_euclid,dt_k1_xboole_0,dt_k5_euclid,dt_k5_ordinal2,dt_l1_metric_1,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_struct_0,fc2_euclid,fc2_finseq_1,fc2_topreal1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_metric_1,rc1_xreal_0,rc2_metric_1,rc2_xreal_0,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc5_struct_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,d8_euclid,existence_m1_finseq_2,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k16_euclid,dt_k1_numbers,dt_k1_zfmisc_1,dt_k4_finseq_2,dt_k5_numbers,dt_m1_finseq_2,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_xreal_0,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_euclid,fc1_subset_1,fc2_membered,rc1_subset_1,rc2_subset_1,t1_real,t2_subset,t3_subset,t4_real,t4_subset,t5_subset,t6_boole,t8_boole,d7_euclid,d9_euclid,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,dt_k1_euclid,dt_c1_46__jordan2c,dt_c2_46__jordan2c,dt_c5_46__jordan2c,de_c5_46__jordan2c,rqLessOrEqual__r1_xreal_0__r1_r1,t1_subset,t7_boole,d1_euclid,spc1_numerals,spc1_boole,e1_46__jordan2c]), [interesting(0.8),file(jordan2c,e8_46__jordan2c),[file(jordan2c,e8_46__jordan2c)]]). fof(e6_46__jordan2c,plain,( r2_hidden(c4_46__jordan2c,c2_46__jordan2c) ), inference(mizar_by,[status(thm),assumptions([dt_c1_46__jordan2c,dt_c2_46__jordan2c,e2_46__jordan2c,e1_46__jordan2c])],[existence_m1_finseq_1,dt_m1_finseq_1,rc2_goboard1,rc4_finseq_1,free_g1_pre_topc,existence_m2_finseq_1,redefinition_m2_finseq_1,dt_g1_pre_topc,dt_k2_finseq_2,dt_m2_finseq_1,dt_u1_pre_topc,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_m1_relset_1,existence_m2_finseq_2,existence_m2_relset_1,redefinition_m2_finseq_2,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k3_jordan2c,dt_k5_pcomps_1,dt_l1_pre_topc,dt_m1_relset_1,dt_m2_finseq_2,dt_m2_relset_1,dt_u1_metric_1,cc1_arytm_3,cc1_finseq_1,cc1_relset_1,cc2_arytm_3,cc2_funct_1,fc14_finset_1,fc3_pcomps_1,fc4_pcomps_1,fc4_subset_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc1_pre_topc,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_pcomps_1,rc2_pre_topc,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,d7_jordan2c,free_g1_metric_1,commutativity_k2_xcmplx_0,reflexivity_r1_tarski,abstractness_v1_metric_1,existence_l1_metric_1,existence_l1_struct_0,redefinition_k4_jordan2c,dt_g1_metric_1,dt_k13_euclid,dt_k15_euclid,dt_k1_xboole_0,dt_k2_xcmplx_0,dt_k4_jordan2c,dt_k5_ordinal2,dt_l1_metric_1,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_struct_0,fc2_euclid,fc2_finseq_1,fc2_topreal1,fc3_xreal_0,fc5_membered,fc6_membered,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_finset_1,rc1_membered,rc1_metric_1,rc1_xreal_0,rc2_metric_1,rc2_xreal_0,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc5_struct_0,spc6_arithm,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,d8_euclid,commutativity_k9_binop_2,existence_m1_finseq_2,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_k9_binop_2,redefinition_m2_subset_1,dt_k14_euclid,dt_k18_euclid,dt_k1_numbers,dt_k1_zfmisc_1,dt_k4_finseq_2,dt_k5_jordan2c,dt_k5_numbers,dt_k9_binop_2,dt_m1_finseq_2,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c3_46__jordan2c,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_xreal_0,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_euclid,fc1_subset_1,fc2_membered,rc1_subset_1,rc2_subset_1,t1_real,t2_subset,t3_subset,t4_real,t4_subset,t5_subset,t6_boole,t8_boole,d7_euclid,d8_jordan2c,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,dt_k1_euclid,dt_c1_46__jordan2c,dt_c2_46__jordan2c,dt_c4_46__jordan2c,de_c4_46__jordan2c,rqLessOrEqual__r1_xreal_0__r1_r1,t1_subset,t7_boole,d1_euclid,spc1_numerals,spc1_boole,e1_46__jordan2c]), [interesting(0.8),file(jordan2c,e6_46__jordan2c),[file(jordan2c,e6_46__jordan2c)]]). fof(e9_46__jordan2c,plain,( r1_xreal_0(k4_metric_1(k14_euclid(c1_46__jordan2c),c4_46__jordan2c,c5_46__jordan2c),c3_46__jordan2c) ), inference(mizar_by,[status(thm),assumptions([dt_c1_46__jordan2c,dt_c2_46__jordan2c,e2_46__jordan2c,e1_46__jordan2c])],[existence_m1_finseq_1,dt_m1_finseq_1,rc2_goboard1,rc4_finseq_1,free_g1_pre_topc,existence_m2_finseq_1,redefinition_k4_finseqop,redefinition_m2_finseq_1,dt_g1_pre_topc,dt_k2_finseq_2,dt_k4_finseqop,dt_m2_finseq_1,dt_u1_pre_topc,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_m2_finseq_2,redefinition_m2_finseq_2,dt_k3_jordan2c,dt_k4_euclid,dt_k5_pcomps_1,dt_l1_pre_topc,dt_m2_finseq_2,cc1_arytm_3,cc1_finseq_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,fc3_pcomps_1,fc4_pcomps_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc1_pre_topc,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_pcomps_1,rc2_pre_topc,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,d7_jordan2c,d4_euclid,commutativity_k2_xcmplx_0,reflexivity_r1_tarski,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_k4_jordan2c,redefinition_k5_euclid,redefinition_m2_relset_1,dt_k15_euclid,dt_k1_xboole_0,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_k4_jordan2c,dt_k5_euclid,dt_k5_ordinal2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_finset_1,fc1_struct_0,fc2_euclid,fc2_finseq_1,fc2_topreal1,fc3_xreal_0,fc4_subset_1,fc5_membered,fc6_membered,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_finset_1,rc1_membered,rc1_xreal_0,rc2_metric_1,rc2_xreal_0,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc5_struct_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,spc6_arithm,t1_arithm,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,d8_euclid,free_g1_metric_1,commutativity_k9_binop_2,abstractness_v1_metric_1,existence_l1_metric_1,existence_l1_struct_0,existence_m2_subset_1,redefinition_k5_numbers,redefinition_k9_binop_2,redefinition_m2_subset_1,dt_g1_metric_1,dt_k13_euclid,dt_k16_euclid,dt_k18_euclid,dt_k1_euclid,dt_k1_numbers,dt_k1_zfmisc_1,dt_k2_metric_1,dt_k5_jordan2c,dt_k5_numbers,dt_k9_binop_2,dt_l1_metric_1,dt_l1_struct_0,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_xreal_0,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_euclid,fc1_subset_1,fc2_membered,rc1_metric_1,rc1_subset_1,rc2_subset_1,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,spc1_boole,spc1_numerals,t1_numerals,t1_real,t2_subset,t3_subset,t4_real,t4_subset,t5_subset,t6_boole,t8_boole,d9_euclid,d1_euclid,d8_jordan2c,spc1_numerals,spc1_boole,commutativity_k4_metric_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,redefinition_k4_metric_1,dt_k14_euclid,dt_k4_metric_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_46__jordan2c,dt_c2_46__jordan2c,dt_c3_46__jordan2c,dt_c4_46__jordan2c,dt_c5_46__jordan2c,de_c4_46__jordan2c,de_c5_46__jordan2c,rqLessOrEqual__r1_xreal_0__r0_r0,t1_subset,t7_boole,d7_euclid,spc0_numerals,spc0_boole,e8_46__jordan2c,e4_46__jordan2c,e6_46__jordan2c]), [interesting(0.8),file(jordan2c,e9_46__jordan2c),[file(jordan2c,e9_46__jordan2c)]]). fof(t45_jgraph_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k14_euclid(A))) => ! [E] : ( m1_subset_1(E,u1_struct_0(k14_euclid(A))) => ( ( D = B & E = C ) => k5_toprns_1(A,k20_euclid(A,B,C)) = k4_metric_1(k14_euclid(A),D,E) ) ) ) ) ) ) ), file(jgraph_1,t45_jgraph_1), [interesting(0.9),axiom,file(jgraph_1,t45_jgraph_1)]). fof(e10_46__jordan2c,plain,( r1_xreal_0(k5_toprns_1(c1_46__jordan2c,k20_euclid(c1_46__jordan2c,k18_euclid(k9_binop_2(c3_46__jordan2c,1),c1_46__jordan2c,k5_jordan2c(c1_46__jordan2c)),k16_euclid(c1_46__jordan2c))),c3_46__jordan2c) ), inference(mizar_by,[status(thm),assumptions([dt_c1_46__jordan2c,dt_c2_46__jordan2c,e2_46__jordan2c,e1_46__jordan2c])],[existence_m1_finseq_1,dt_m1_finseq_1,rc2_goboard1,rc4_finseq_1,existence_m2_finseq_1,redefinition_k4_finseqop,redefinition_m2_finseq_1,dt_k2_finseq_2,dt_k4_finseqop,dt_m2_finseq_1,cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,spc0_boole,spc0_numerals,t1_arithm,t1_numerals,spc0_numerals,spc0_boole,free_g1_pre_topc,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_finseq_2,existence_m2_relset_1,redefinition_m2_finseq_2,redefinition_m2_relset_1,dt_g1_pre_topc,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k3_jordan2c,dt_k4_euclid,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_finseq_2,dt_m2_relset_1,dt_u1_metric_1,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_finset_1,fc2_finseq_1,fc4_subset_1,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_arytm_3,rc1_finset_1,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,spc6_arithm,t1_subset,t2_real,t3_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,d7_jordan2c,d4_euclid,free_g1_metric_1,commutativity_k2_xcmplx_0,abstractness_v1_metric_1,abstractness_v1_pre_topc,existence_l1_metric_1,existence_l1_pre_topc,existence_l1_struct_0,redefinition_k4_jordan2c,redefinition_k5_euclid,dt_g1_metric_1,dt_k13_euclid,dt_k1_euclid,dt_k1_zfmisc_1,dt_k2_metric_1,dt_k2_xcmplx_0,dt_k4_jordan2c,dt_k5_euclid,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_metric_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_struct_0,fc1_subset_1,fc3_pcomps_1,fc3_xreal_0,fc4_pcomps_1,fc5_membered,fc8_xreal_0,rc1_metric_1,rc1_pre_topc,rc1_subset_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc3_metric_1,rc3_struct_0,rc5_struct_0,t1_real,t2_subset,t3_subset,t4_real,t6_boole,t7_boole,t8_boole,d1_euclid,commutativity_k4_metric_1,commutativity_k9_binop_2,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k4_metric_1,redefinition_k5_numbers,redefinition_k9_binop_2,redefinition_m2_subset_1,dt_k14_euclid,dt_k15_euclid,dt_k16_euclid,dt_k18_euclid,dt_k1_numbers,dt_k20_euclid,dt_k4_metric_1,dt_k5_jordan2c,dt_k5_numbers,dt_k5_toprns_1,dt_k9_binop_2,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_46__jordan2c,dt_c3_46__jordan2c,dt_c4_46__jordan2c,dt_c5_46__jordan2c,de_c4_46__jordan2c,de_c5_46__jordan2c,fc1_euclid,fc2_euclid,fc2_membered,fc2_topreal1,rqLessOrEqual__r1_xreal_0__r1_r1,d7_euclid,d8_euclid,d9_euclid,d8_jordan2c,spc1_numerals,spc1_boole,e9_46__jordan2c,t45_jgraph_1]), [interesting(0.8),file(jordan2c,e10_46__jordan2c),[file(jordan2c,e10_46__jordan2c)]]). fof(t13_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ( k20_euclid(A,B,k16_euclid(A)) = B & k20_euclid(A,k16_euclid(A),B) = k19_euclid(A,B) ) ) ) ), file(jordan2c,t13_jordan2c), [interesting(0.9),axiom,file(jordan2c,t13_jordan2c)]). fof(e11_46__jordan2c,plain,( r1_xreal_0(k5_toprns_1(c1_46__jordan2c,k18_euclid(k9_binop_2(c3_46__jordan2c,1),c1_46__jordan2c,k5_jordan2c(c1_46__jordan2c))),c3_46__jordan2c) ), inference(mizar_by,[status(thm),assumptions([dt_c1_46__jordan2c,dt_c2_46__jordan2c,e2_46__jordan2c,e1_46__jordan2c])],[existence_m1_finseq_1,dt_m1_finseq_1,rc2_goboard1,rc4_finseq_1,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_finseq_1,existence_m2_relset_1,redefinition_k4_finseqop,redefinition_m2_finseq_1,redefinition_m2_relset_1,dt_k2_finseq_2,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_k4_finseqop,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_finseq_1,dt_m2_relset_1,dt_u1_metric_1,cc1_finseq_1,cc1_relset_1,cc2_funct_1,fc14_finset_1,fc4_subset_1,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,spc0_boole,spc0_numerals,t1_arithm,t1_numerals,spc0_numerals,spc0_boole,free_g1_metric_1,free_g1_pre_topc,reflexivity_r1_tarski,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,existence_m2_finseq_2,redefinition_m2_finseq_2,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k3_jordan2c,dt_k4_euclid,dt_l1_metric_1,dt_m2_finseq_2,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc2_finseq_1,fc3_pcomps_1,fc4_pcomps_1,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_arytm_3,rc1_finset_1,rc1_membered,rc1_metric_1,rc1_xreal_0,rc2_metric_1,rc2_xreal_0,rc3_finset_1,rc3_metric_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,spc6_arithm,t1_subset,t2_real,t3_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,d1_euclid,d7_jordan2c,d4_euclid,commutativity_k2_xcmplx_0,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,redefinition_k4_jordan2c,redefinition_k5_euclid,dt_k14_euclid,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k4_jordan2c,dt_k5_euclid,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_euclid,fc1_struct_0,fc1_subset_1,fc3_xreal_0,fc5_membered,fc8_xreal_0,rc1_pre_topc,rc1_subset_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc3_struct_0,rc5_struct_0,t1_real,t2_subset,t3_subset,t4_real,t6_boole,t7_boole,t8_boole,d7_euclid,commutativity_k9_binop_2,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_k9_binop_2,redefinition_m2_subset_1,dt_k15_euclid,dt_k16_euclid,dt_k18_euclid,dt_k19_euclid,dt_k1_numbers,dt_k20_euclid,dt_k5_jordan2c,dt_k5_numbers,dt_k5_toprns_1,dt_k9_binop_2,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_46__jordan2c,dt_c3_46__jordan2c,fc2_euclid,fc2_membered,fc2_topreal1,rqLessOrEqual__r1_xreal_0__r1_r1,d8_euclid,d9_euclid,d8_jordan2c,spc1_numerals,spc1_boole,e10_46__jordan2c,t13_jordan2c]), [interesting(0.8),file(jordan2c,e11_46__jordan2c),[file(jordan2c,e11_46__jordan2c)]]). fof(t8_toprns_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => k4_real_1(k18_complex1(B),k5_toprns_1(A,C)) = k5_toprns_1(A,k18_euclid(B,A,C)) ) ) ) ), file(toprns_1,t8_toprns_1), [interesting(0.9),axiom,file(toprns_1,t8_toprns_1)]). fof(e12_46__jordan2c,plain,( r1_xreal_0(k11_binop_2(k18_complex1(k9_binop_2(c3_46__jordan2c,1)),k5_toprns_1(c1_46__jordan2c,k5_jordan2c(c1_46__jordan2c))),c3_46__jordan2c) ), inference(mizar_by,[status(thm),assumptions([dt_c1_46__jordan2c,dt_c2_46__jordan2c,e2_46__jordan2c,e1_46__jordan2c])],[existence_m1_finseq_1,dt_m1_finseq_1,rc2_goboard1,rc4_finseq_1,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_finseq_1,existence_m2_relset_1,redefinition_m2_finseq_1,redefinition_m2_relset_1,dt_k2_finseq_2,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_finseq_1,dt_m2_relset_1,dt_u1_metric_1,cc1_finseq_1,cc1_relset_1,cc2_funct_1,fc14_finset_1,fc4_subset_1,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,free_g1_pre_topc,reflexivity_r1_tarski,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,existence_m2_finseq_2,redefinition_m2_finseq_2,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k3_jordan2c,dt_l1_metric_1,dt_m2_finseq_2,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_finseq_1,fc3_pcomps_1,fc4_pcomps_1,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_arytm_3,rc1_finset_1,rc1_membered,rc1_metric_1,rc2_metric_1,rc2_xreal_0,rc3_finset_1,rc3_metric_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_subset,t2_real,t3_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,d1_euclid,d7_jordan2c,projectivity_k16_complex1,commutativity_k2_xcmplx_0,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,redefinition_k4_jordan2c,dt_k14_euclid,dt_k16_complex1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k4_jordan2c,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_euclid,fc1_struct_0,fc1_subset_1,fc23_xreal_0,fc3_xreal_0,fc4_xreal_0,fc5_membered,fc8_xreal_0,rc1_pre_topc,rc1_subset_1,rc1_xreal_0,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc3_struct_0,rc5_struct_0,spc5_arithm,spc6_arithm,spc7_arithm,t1_real,t2_subset,t3_arithm,t3_subset,t4_real,t6_boole,t7_boole,t8_boole,d7_euclid,commutativity_k11_binop_2,projectivity_k18_complex1,commutativity_k3_xcmplx_0,commutativity_k4_real_1,commutativity_k9_binop_2,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k11_binop_2,redefinition_k18_complex1,redefinition_k4_real_1,redefinition_k5_numbers,redefinition_k9_binop_2,redefinition_m2_subset_1,dt_k11_binop_2,dt_k15_euclid,dt_k18_complex1,dt_k18_euclid,dt_k1_numbers,dt_k3_xcmplx_0,dt_k4_real_1,dt_k5_jordan2c,dt_k5_numbers,dt_k5_toprns_1,dt_k9_binop_2,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_46__jordan2c,dt_c3_46__jordan2c,fc2_euclid,fc2_membered,fc2_topreal1,rqLessOrEqual__r1_xreal_0__r1_r1,d8_euclid,d8_jordan2c,spc1_numerals,spc1_boole,e11_46__jordan2c,t8_toprns_1,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.8),file(jordan2c,e12_46__jordan2c),[file(jordan2c,e12_46__jordan2c)]]). fof(t37_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k5_toprns_1(A,k5_jordan2c(A)) = k9_square_1(A) ) ), file(jordan2c,t37_jordan2c), [interesting(0.9),axiom,file(jordan2c,t37_jordan2c)]). fof(e13_46__jordan2c,plain,( r1_xreal_0(k11_binop_2(k18_complex1(k9_binop_2(c3_46__jordan2c,1)),k9_square_1(c1_46__jordan2c)),c3_46__jordan2c) ), inference(mizar_by,[status(thm),assumptions([dt_c1_46__jordan2c,dt_c2_46__jordan2c,e2_46__jordan2c,e1_46__jordan2c])],[existence_m1_finseq_1,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_m1_finseq_1,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_relset_1,fc14_finset_1,fc4_subset_1,rc1_jordan2c,rc2_finseq_1,rc2_goboard1,rc2_tbsp_1,rc3_tbsp_1,rc4_finseq_1,free_g1_metric_1,free_g1_pre_topc,abstractness_v1_metric_1,existence_l1_metric_1,existence_m1_finseq_2,existence_m2_finseq_1,redefinition_m2_finseq_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k2_finseq_2,dt_k4_finseq_2,dt_l1_metric_1,dt_m1_finseq_2,dt_m2_finseq_1,dt_u1_pre_topc,cc1_finseq_1,cc2_funct_1,cc2_tbsp_1,fc3_pcomps_1,fc4_pcomps_1,rc1_finseq_1,rc1_funct_1,rc1_metric_1,rc2_finset_1,rc2_funct_1,rc2_metric_1,rc3_finseq_1,rc3_funct_1,rc3_metric_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m2_finseq_2,redefinition_m2_finseq_2,dt_k14_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k3_jordan2c,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_finseq_2,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_finset_1,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_euclid,fc1_struct_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_finseq_1,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_arytm_3,rc1_finset_1,rc1_membered,rc1_pre_topc,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_finset_1,rc3_struct_0,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc5_struct_0,t1_subset,t2_real,t3_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,d7_euclid,d1_euclid,d7_jordan2c,projectivity_k16_complex1,commutativity_k2_xcmplx_0,existence_m1_subset_1,redefinition_k4_jordan2c,dt_k15_euclid,dt_k16_complex1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k4_jordan2c,dt_k5_ordinal2,dt_k8_square_1,dt_m1_subset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_subset_1,fc23_xreal_0,fc2_euclid,fc2_topreal1,fc3_xreal_0,fc4_xreal_0,fc5_membered,fc8_xreal_0,rc1_subset_1,rc1_xreal_0,rc2_subset_1,spc5_arithm,spc6_arithm,spc7_arithm,t1_real,t2_subset,t3_arithm,t3_subset,t4_real,t6_boole,t7_boole,t8_boole,d8_euclid,commutativity_k11_binop_2,projectivity_k18_complex1,commutativity_k3_xcmplx_0,commutativity_k9_binop_2,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m2_subset_1,redefinition_k11_binop_2,redefinition_k18_complex1,redefinition_k5_numbers,redefinition_k9_binop_2,redefinition_k9_square_1,redefinition_m2_subset_1,dt_k11_binop_2,dt_k18_complex1,dt_k1_numbers,dt_k3_xcmplx_0,dt_k5_jordan2c,dt_k5_numbers,dt_k5_toprns_1,dt_k9_binop_2,dt_k9_square_1,dt_m2_subset_1,dt_c1_46__jordan2c,dt_c3_46__jordan2c,fc2_membered,rqLessOrEqual__r1_xreal_0__r1_r1,d8_jordan2c,spc1_numerals,spc1_boole,e12_46__jordan2c,t37_jordan2c,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.8),file(jordan2c,e13_46__jordan2c),[file(jordan2c,e13_46__jordan2c)]]). fof(d1_absvalue,definition,( ! [A] : ( v1_xreal_0(A) => ( ( r1_xreal_0(0,A) => k16_complex1(A) = A ) & ( ~ r1_xreal_0(0,A) => k16_complex1(A) = k4_xcmplx_0(A) ) ) ) ), file(absvalue,d1_absvalue), [interesting(0.9),axiom,file(absvalue,d1_absvalue)]). fof(e16_46__jordan2c,plain,( r1_xreal_0(k11_binop_2(k9_binop_2(c3_46__jordan2c,1),k9_square_1(c1_46__jordan2c)),c3_46__jordan2c) ), inference(mizar_by,[status(thm),assumptions([dt_c1_46__jordan2c,dt_c2_46__jordan2c,e2_46__jordan2c,e1_46__jordan2c])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_finseq_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_xreal_0,fc1_subset_1,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_finseq_1,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_finset_1,rc1_membered,rc1_subset_1,rc2_subset_1,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_xcmplx_0,dt_k5_numbers,dt_k8_square_1,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc23_xreal_0,fc2_membered,fc3_xreal_0,fc8_xreal_0,rc1_xreal_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0,spc2_arithm,spc5_arithm,spc6_arithm,spc7_arithm,spc8_arithm,t1_arithm,t1_numerals,t1_real,t2_arithm,t2_subset,t3_arithm,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k11_binop_2,projectivity_k16_complex1,projectivity_k18_complex1,commutativity_k3_xcmplx_0,involutiveness_k4_xcmplx_0,commutativity_k9_binop_2,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k11_binop_2,redefinition_k18_complex1,redefinition_k9_binop_2,redefinition_k9_square_1,dt_k11_binop_2,dt_k16_complex1,dt_k18_complex1,dt_k3_xcmplx_0,dt_k4_xcmplx_0,dt_k9_binop_2,dt_k9_square_1,dt_c1_46__jordan2c,dt_c3_46__jordan2c,cc2_xreal_0,fc1_xreal_0,fc4_xreal_0,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealNeg__k4_xcmplx_0__rm1_r1,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e15_46__jordan2c,e13_46__jordan2c,d1_absvalue,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1]), [interesting(0.8),file(jordan2c,e16_46__jordan2c),[file(jordan2c,e16_46__jordan2c)]]). fof(t2_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ( r1_xreal_0(A,B) & r1_xreal_0(B,C) ) => r1_xreal_0(A,C) ) ) ) ) ), file(xreal_1,t2_xreal_1), [interesting(0.9),axiom,file(xreal_1,t2_xreal_1)]). fof(e19_46__jordan2c,plain,( r1_xreal_0(k11_binop_2(k9_binop_2(c3_46__jordan2c,1),1),c3_46__jordan2c) ), inference(mizar_by,[status(thm),assumptions([dt_c1_46__jordan2c,dt_c2_46__jordan2c,e2_46__jordan2c,e1_46__jordan2c])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_finseq_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_subset_1,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_finseq_1,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_finset_1,rc1_membered,rc1_subset_1,rc2_subset_1,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_xcmplx_0,dt_k5_numbers,dt_k8_square_1,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc23_xreal_0,fc2_membered,fc3_xreal_0,fc8_xreal_0,rc1_xreal_0,spc5_arithm,spc6_arithm,spc7_arithm,t1_real,t2_subset,t3_arithm,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k11_binop_2,commutativity_k3_xcmplx_0,commutativity_k9_binop_2,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k11_binop_2,redefinition_k9_binop_2,redefinition_k9_square_1,dt_k11_binop_2,dt_k3_xcmplx_0,dt_k9_binop_2,dt_k9_square_1,dt_c1_46__jordan2c,dt_c3_46__jordan2c,cc2_xreal_0,fc4_xreal_0,spc1_numerals,spc1_boole,e18_46__jordan2c,e16_46__jordan2c,t2_xreal_1,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.8),file(jordan2c,e19_46__jordan2c),[file(jordan2c,e19_46__jordan2c)]]). fof(t11_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( r1_xreal_0(A,B) <=> r1_xreal_0(k6_xcmplx_0(A,C),k6_xcmplx_0(B,C)) ) ) ) ) ), file(xreal_1,t11_xreal_1), [interesting(0.9),axiom,file(xreal_1,t11_xreal_1)]). fof(e20_46__jordan2c,plain,( r1_xreal_0(k10_binop_2(k9_binop_2(c3_46__jordan2c,1),c3_46__jordan2c),k10_binop_2(c3_46__jordan2c,c3_46__jordan2c)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_46__jordan2c,dt_c2_46__jordan2c,e2_46__jordan2c,e1_46__jordan2c])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_finseq_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_subset_1,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_finseq_1,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_finset_1,rc1_membered,rc1_subset_1,rc2_subset_1,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc23_xreal_0,fc2_membered,fc8_xreal_0,rc1_xreal_0,spc1_arithm,spc2_arithm,spc5_arithm,spc6_arithm,spc7_arithm,spc8_arithm,spc9_arithm,t1_arithm,t1_numerals,t1_real,t2_arithm,t2_subset,t3_arithm,t4_arithm,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k11_binop_2,commutativity_k2_xcmplx_0,commutativity_k3_xcmplx_0,involutiveness_k4_xcmplx_0,commutativity_k9_binop_2,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k10_binop_2,redefinition_k11_binop_2,redefinition_k9_binop_2,dt_k10_binop_2,dt_k11_binop_2,dt_k2_xcmplx_0,dt_k3_xcmplx_0,dt_k4_xcmplx_0,dt_k6_xcmplx_0,dt_k9_binop_2,dt_c3_46__jordan2c,cc2_xreal_0,fc1_xreal_0,fc3_xreal_0,fc4_xreal_0,fc5_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e19_46__jordan2c,t11_xreal_1,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.8),file(jordan2c,e20_46__jordan2c),[file(jordan2c,e20_46__jordan2c)]]). fof(e21_46__jordan2c,plain,( r1_xreal_0(1,0) ), inference(mizar_by,[status(thm),assumptions([dt_c1_46__jordan2c,dt_c2_46__jordan2c,e2_46__jordan2c,e1_46__jordan2c])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_finseq_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_subset_1,fc20_xreal_0,fc2_finseq_1,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_finset_1,rc1_membered,rc1_subset_1,rc2_subset_1,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_k6_xcmplx_0,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_xreal_0,fc2_membered,fc3_xreal_0,fc5_xreal_0,fc8_xreal_0,rc1_xreal_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,spc1_arithm,spc6_arithm,spc8_arithm,spc9_arithm,t1_arithm,t1_numerals,t1_real,t2_subset,t4_arithm,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k2_xcmplx_0,involutiveness_k4_xcmplx_0,commutativity_k9_binop_2,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k10_binop_2,redefinition_k9_binop_2,dt_k10_binop_2,dt_k2_xcmplx_0,dt_k4_xcmplx_0,dt_k9_binop_2,dt_c3_46__jordan2c,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e20_46__jordan2c,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__rm1_r1_r0]), [interesting(0.8),file(jordan2c,e21_46__jordan2c),[file(jordan2c,e21_46__jordan2c)]]). fof(e22_46__jordan2c,plain,( ~ $true ), inference(mizar_by,[status(thm),assumptions([dt_c1_46__jordan2c,dt_c2_46__jordan2c,e2_46__jordan2c,e1_46__jordan2c])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_finseq_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc1_subset_1,fc2_finseq_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_subset_1,rc1_xreal_0,rc2_subset_1,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_boole,t8_real,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc2_membered,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t7_boole,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e21_46__jordan2c]), [interesting(0.8),file(jordan2c,e22_46__jordan2c),[file(jordan2c,e22_46__jordan2c)]]). fof(i5_46__jordan2c,theorem,( $true ), introduced(tautology,[file(jordan2c,i5_46__jordan2c)]), [interesting(0.8),trivial,file(jordan2c,i5_46__jordan2c)]). fof(i4_46__jordan2c,plain,( ~ $true ), inference(conclusion,[status(thm),assumptions([dt_c1_46__jordan2c,dt_c2_46__jordan2c,e2_46__jordan2c,e1_46__jordan2c])],[e22_46__jordan2c,i5_46__jordan2c]), [interesting(0.8),file(jordan2c,i4_46__jordan2c),[file(jordan2c,i4_46__jordan2c)]]). fof(i3_46__jordan2c,plain,( ~ v6_tbsp_1(c2_46__jordan2c,k14_euclid(c1_46__jordan2c)) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_46__jordan2c,dt_c2_46__jordan2c,e1_46__jordan2c]),discharge_asm(discharge,[e2_46__jordan2c])],[e2_46__jordan2c,i4_46__jordan2c]), [interesting(0.8),file(jordan2c,i3_46__jordan2c),[file(jordan2c,i3_46__jordan2c)]]). fof(i2_46__jordan2c,plain,( ~ ( r1_xreal_0(1,c1_46__jordan2c) & c2_46__jordan2c = k1_euclid(c1_46__jordan2c) & v6_tbsp_1(c2_46__jordan2c,k14_euclid(c1_46__jordan2c)) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_46__jordan2c,dt_c2_46__jordan2c]),discharge_asm(discharge,[e1_46__jordan2c])],[e1_46__jordan2c,i3_46__jordan2c]), [interesting(0.8),file(jordan2c,i2_46__jordan2c),[file(jordan2c,i2_46__jordan2c)]]). fof(i2_46_tmp__jordan2c,plain, ( m1_subset_1(c2_46__jordan2c,k1_zfmisc_1(u1_struct_0(k14_euclid(c1_46__jordan2c)))) => ~ ( r1_xreal_0(1,c1_46__jordan2c) & c2_46__jordan2c = k1_euclid(c1_46__jordan2c) & v6_tbsp_1(c2_46__jordan2c,k14_euclid(c1_46__jordan2c)) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_46__jordan2c]),discharge_asm(discharge,[dt_c2_46__jordan2c])],[dt_c2_46__jordan2c,i2_46__jordan2c]), [interesting(0.8),i1_46__jordan2c]). fof(i1_46__jordan2c,plain,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k14_euclid(c1_46__jordan2c)))) => ~ ( r1_xreal_0(1,c1_46__jordan2c) & A = k1_euclid(c1_46__jordan2c) & v6_tbsp_1(A,k14_euclid(c1_46__jordan2c)) ) ) ), inference(let,[status(thm),assumptions([dt_c1_46__jordan2c])],[i2_46_tmp__jordan2c,dh_c2_46__jordan2c]), [interesting(0.8),file(jordan2c,i1_46__jordan2c),[file(jordan2c,i1_46__jordan2c)]]). fof(i1_46_tmp__jordan2c,plain, ( m2_subset_1(c1_46__jordan2c,k1_numbers,k5_numbers) => ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k14_euclid(c1_46__jordan2c)))) => ~ ( r1_xreal_0(1,c1_46__jordan2c) & A = k1_euclid(c1_46__jordan2c) & v6_tbsp_1(A,k14_euclid(c1_46__jordan2c)) ) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_46__jordan2c])],[dt_c1_46__jordan2c,i1_46__jordan2c]), [interesting(1),t39_jordan2c]). fof(t39_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k14_euclid(A)))) => ~ ( r1_xreal_0(1,A) & B = k1_euclid(A) & v6_tbsp_1(B,k14_euclid(A)) ) ) ) ), inference(let,[status(thm),assumptions([])],[i1_46_tmp__jordan2c,dh_c1_46__jordan2c]), [interesting(1),file(jordan2c,t39_jordan2c),[file(jordan2c,t39_jordan2c)]]).