% Mizar ND problem: t5_newton,newton,52,61 fof(dh_c1_2__newton,definition, ( ( m2_subset_1(c1_2__newton,k1_numbers,k5_numbers) => ( r1_xreal_0(1,c1_2__newton) => k2_finseq_1(c1_2__newton) = k2_xboole_0(k2_xboole_0(k1_tarski(1),a_1_0_newton(c1_2__newton)),k1_tarski(c1_2__newton)) ) ) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r1_xreal_0(1,A) => k2_finseq_1(A) = k2_xboole_0(k2_xboole_0(k1_tarski(1),a_1_0_newton(A)),k1_tarski(A)) ) ) ), introduced(definition,[new_symbol(c1_2__newton),file(newton,c1_2__newton)]), [interesting(0.8),axiom,file(newton,c1_2__newton)]). fof(e1_2__newton,assumption,( r1_xreal_0(1,c1_2__newton) ), introduced(assumption,[file(newton,e1_2__newton)]), [interesting(0.8),axiom,file(newton,e1_2__newton)]). 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(fc9_membered,theorem,( ! [A] : ( v1_rat_1(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) & v3_membered(k1_tarski(A)) ) ) ), file(membered,fc9_membered), [interesting(0.9),axiom,file(membered,fc9_membered)]). 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(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(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(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(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]). 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_membered,theorem,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), [interesting(0.9),axiom,file(membered,cc2_membered)]). 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_membered,theorem,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), [interesting(0.9),axiom,file(membered,cc3_membered)]). 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(cc4_int_1,theorem,( ! [A] : ( v1_int_1(A) => ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(int_1,cc4_int_1), [interesting(0.9),axiom,file(int_1,cc4_int_1)]). 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_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(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(fc10_membered,theorem,( ! [A] : ( v1_int_1(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) & v3_membered(k1_tarski(A)) & v4_membered(k1_tarski(A)) ) ) ), file(membered,fc10_membered), [interesting(0.9),axiom,file(membered,fc10_membered)]). fof(fc1_ordinal2,theorem, ( v1_ordinal1(k5_ordinal2) & v2_ordinal1(k5_ordinal2) & v3_ordinal1(k5_ordinal2) & ~ v1_xboole_0(k5_ordinal2) ), file(ordinal2,fc1_ordinal2), [interesting(0.9),axiom,file(ordinal2,fc1_ordinal2)]). fof(fc22_membered,theorem,( ! [A,B] : ( ( v1_membered(A) & v1_membered(B) ) => v1_membered(k2_xboole_0(A,B)) ) ), file(membered,fc22_membered), [interesting(0.9),axiom,file(membered,fc22_membered)]). fof(fc23_membered,theorem,( ! [A,B] : ( ( v2_membered(A) & v2_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc23_membered), [interesting(0.9),axiom,file(membered,fc23_membered)]). fof(fc24_membered,theorem,( ! [A,B] : ( ( v3_membered(A) & v3_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) & v3_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc24_membered), [interesting(0.9),axiom,file(membered,fc24_membered)]). fof(fc25_membered,theorem,( ! [A,B] : ( ( v4_membered(A) & v4_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) & v3_membered(k2_xboole_0(A,B)) & v4_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc25_membered), [interesting(0.9),axiom,file(membered,fc25_membered)]). fof(fc26_membered,theorem,( ! [A,B] : ( ( v5_membered(A) & v5_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) & v3_membered(k2_xboole_0(A,B)) & v4_membered(k2_xboole_0(A,B)) & v5_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc26_membered), [interesting(0.9),axiom,file(membered,fc26_membered)]). 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(fc7_membered,theorem,( ! [A] : ( v1_xcmplx_0(A) => v1_membered(k1_tarski(A)) ) ), file(membered,fc7_membered), [interesting(0.9),axiom,file(membered,fc7_membered)]). fof(fc8_membered,theorem,( ! [A] : ( v1_xreal_0(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) ) ) ), file(membered,fc8_membered), [interesting(0.9),axiom,file(membered,fc8_membered)]). fof(rc1_int_1,theorem,( ? [A] : ( m1_subset_1(A,k1_numbers) & v1_xcmplx_0(A) & v1_xreal_0(A) & v1_int_1(A) ) ), file(int_1,rc1_int_1), [interesting(0.9),axiom,file(int_1,rc1_int_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_nat_1,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc1_nat_1), [interesting(0.9),axiom,file(nat_1,rc1_nat_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_int_1,theorem,( ? [A] : v1_int_1(A) ), file(int_1,rc2_int_1), [interesting(0.9),axiom,file(int_1,rc2_int_1)]). fof(rc2_nat_1,theorem,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(A) & v3_ordinal1(A) ) ), file(nat_1,rc2_nat_1), [interesting(0.9),axiom,file(nat_1,rc2_nat_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_nat_1,theorem,( ? [A] : ( m1_subset_1(A,k5_numbers) & ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc3_nat_1), [interesting(0.9),axiom,file(nat_1,rc3_nat_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_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_boole,theorem,( ! [A] : k2_xboole_0(A,k1_xboole_0) = A ), file(boole,t1_boole), [interesting(0.9),axiom,file(boole,t1_boole)]). 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_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(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(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(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_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(fraenkel_a_1_0_finseq_1,definition,( ! [A,B] : ( v4_ordinal2(B) => ( r2_hidden(A,a_1_0_finseq_1(B)) <=> ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & A = C & r1_xreal_0(1,C) & r1_xreal_0(C,B) ) ) ) ), file(finseq_1,a_1_0_finseq_1), [interesting(0.9),axiom,file(finseq_1,a_1_0_finseq_1)]). 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(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(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_finseq_1,axiom,( $true ), file(finseq_1,k1_finseq_1), [interesting(0.9),axiom,file(finseq_1,k1_finseq_1)]). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), [interesting(0.9),axiom,file(numbers,k1_numbers)]). 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_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_nat_1,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(nat_1,cc1_nat_1), [interesting(0.9),axiom,file(nat_1,cc1_nat_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_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_int_1,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_int_1(A) ) ) ), file(int_1,cc2_int_1), [interesting(0.9),axiom,file(int_1,cc2_int_1)]). fof(cc2_nat_1,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc2_nat_1), [interesting(0.9),axiom,file(nat_1,cc2_nat_1)]). fof(cc3_int_1,theorem,( ! [A] : ( v4_ordinal2(A) => v1_int_1(A) ) ), file(int_1,cc3_int_1), [interesting(0.9),axiom,file(int_1,cc3_int_1)]). fof(cc3_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => ( v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc3_nat_1), [interesting(0.9),axiom,file(nat_1,cc3_nat_1)]). 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(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(fc11_membered,theorem,( ! [A] : ( v4_ordinal2(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) & v3_membered(k1_tarski(A)) & v4_membered(k1_tarski(A)) & v5_membered(k1_tarski(A)) ) ) ), file(membered,fc11_membered), [interesting(0.9),axiom,file(membered,fc11_membered)]). fof(fc1_finseq_1,theorem,( ! [A] : ( v4_ordinal2(A) => v1_finset_1(k1_finseq_1(A)) ) ), file(finseq_1,fc1_finseq_1), [interesting(0.9),axiom,file(finseq_1,fc1_finseq_1)]). 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(fc9_finset_1,theorem,( ! [A,B] : ( ( v1_finset_1(A) & v1_finset_1(B) ) => v1_finset_1(k2_xboole_0(A,B)) ) ), file(finset_1,fc9_finset_1), [interesting(0.9),axiom,file(finset_1,fc9_finset_1)]). 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(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(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(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(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_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_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_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_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(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(d1_finseq_1,definition,( ! [A] : ( v4_ordinal2(A) => k1_finseq_1(A) = a_1_0_finseq_1(A) ) ), file(finseq_1,d1_finseq_1), [interesting(0.9),axiom,file(finseq_1,d1_finseq_1)]). fof(commutativity_k2_xboole_0,theorem,( ! [A,B] : k2_xboole_0(A,B) = k2_xboole_0(B,A) ), file(xboole_0,k2_xboole_0), [interesting(0.9),axiom,file(xboole_0,k2_xboole_0)]). fof(idempotence_k2_xboole_0,theorem,( ! [A,B] : k2_xboole_0(A,A) = A ), file(xboole_0,k2_xboole_0), [interesting(0.9),axiom,file(xboole_0,k2_xboole_0)]). fof(reflexivity_r1_tarski,theorem,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), [interesting(0.9),axiom,file(tarski,r1_tarski)]). fof(redefinition_k2_finseq_1,definition,( ! [A] : ( v4_ordinal2(A) => k2_finseq_1(A) = k1_finseq_1(A) ) ), file(finseq_1,k2_finseq_1), [interesting(0.9),axiom,file(finseq_1,k2_finseq_1)]). fof(dt_k1_tarski,axiom,( $true ), file(tarski,k1_tarski), [interesting(0.9),axiom,file(tarski,k1_tarski)]). fof(dt_k2_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => m1_subset_1(k2_finseq_1(A),k1_zfmisc_1(k5_numbers)) ) ), file(finseq_1,k2_finseq_1), [interesting(0.9),axiom,file(finseq_1,k2_finseq_1)]). fof(dt_k2_xboole_0,axiom,( $true ), file(xboole_0,k2_xboole_0), [interesting(0.9),axiom,file(xboole_0,k2_xboole_0)]). fof(dt_c1_2__newton,assumption,( m2_subset_1(c1_2__newton,k1_numbers,k5_numbers) ), introduced(assumption,[file(newton,c1_2__newton)]), [interesting(0.8),axiom,file(newton,c1_2__newton)]). fof(fc1_finset_1,theorem,( ! [A] : ( ~ v1_xboole_0(k1_tarski(A)) & v1_finset_1(k1_tarski(A)) ) ), file(finset_1,fc1_finset_1), [interesting(0.9),axiom,file(finset_1,fc1_finset_1)]). 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(t2_tarski,theorem,( ! [A,B] : ( ! [C] : ( r2_hidden(C,A) <=> r2_hidden(C,B) ) => A = B ) ), file(tarski,t2_tarski), [interesting(0.9),axiom,file(tarski,t2_tarski)]). fof(fraenkel_a_1_0_newton,definition,( ! [A,B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( r2_hidden(A,a_1_0_newton(B)) <=> ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & A = C & ~ r1_xreal_0(C,1) & ~ r1_xreal_0(B,C) ) ) ) ), file(newton,a_1_0_newton), [interesting(0.9),axiom,file(newton,a_1_0_newton)]). 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(spc1_boole,theorem,( ~ v1_xboole_0(1) ), file(boole,spc1_boole), [interesting(0.9),axiom,file(boole,spc1_boole)]). fof(dh_c1_2_2_1__newton,definition, ( ( r2_hidden(c1_2_2_1__newton,k2_xboole_0(k2_xboole_0(k1_tarski(1),a_1_0_newton(c1_2__newton)),k1_tarski(c1_2__newton))) => r2_hidden(c1_2_2_1__newton,k2_finseq_1(c1_2__newton)) ) => ! [A] : ( r2_hidden(A,k2_xboole_0(k2_xboole_0(k1_tarski(1),a_1_0_newton(c1_2__newton)),k1_tarski(c1_2__newton))) => r2_hidden(A,k2_finseq_1(c1_2__newton)) ) ), introduced(definition,[new_symbol(c1_2_2_1__newton),file(newton,c1_2_2_1__newton)]), [interesting(0.5),axiom,file(newton,c1_2_2_1__newton)]). fof(e4_2_2_1__newton,assumption,( r2_hidden(c1_2_2_1__newton,k2_xboole_0(k2_xboole_0(k1_tarski(1),a_1_0_newton(c1_2__newton)),k1_tarski(c1_2__newton))) ), introduced(assumption,[file(newton,e4_2_2_1__newton)]), [interesting(0.5),axiom,file(newton,e4_2_2_1__newton)]). fof(dt_c1_2_2_1__newton,assumption,( $true ), introduced(assumption,[file(newton,c1_2_2_1__newton)]), [interesting(0.5),axiom,file(newton,c1_2_2_1__newton)]). fof(d2_xboole_0,definition,( ! [A,B,C] : ( C = k2_xboole_0(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( r2_hidden(D,A) | r2_hidden(D,B) ) ) ) ), file(xboole_0,d2_xboole_0), [interesting(0.9),axiom,file(xboole_0,d2_xboole_0)]). fof(e5_2_2_1__newton,plain, ( r2_hidden(c1_2_2_1__newton,k2_xboole_0(k1_tarski(1),a_1_0_newton(c1_2__newton))) | r2_hidden(c1_2_2_1__newton,k1_tarski(c1_2__newton)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_2__newton,dt_c1_2_2_1__newton,e4_2_2_1__newton])],[reflexivity_r1_tarski,cc1_finseq_1,fc9_membered,rc1_finseq_1,rc2_finset_1,rc3_finseq_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,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,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_int_1,cc3_membered,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,cc9_membered,fc10_membered,fc11_membered,fc1_ordinal2,fc22_membered,fc23_membered,fc24_membered,fc25_membered,fc26_membered,fc2_finseq_1,fc5_membered,fc6_membered,fc7_membered,fc8_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_boole,t1_real,t2_real,t3_real,t3_subset,t4_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,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_nat_1,cc2_int_1,cc2_nat_1,fc2_membered,fc9_finset_1,rc1_finset_1,rqLessOrEqual__r1_xreal_0__r1_r1,t2_subset,t6_boole,t8_boole,commutativity_k2_xboole_0,idempotence_k2_xboole_0,antisymmetry_r2_hidden,dt_k1_tarski,dt_k2_xboole_0,dt_c1_2__newton,dt_c1_2_2_1__newton,fc1_finset_1,t1_subset,t7_boole,t2_tarski,fraenkel_a_1_0_newton,spc1_numerals,spc1_boole,e4_2_2_1__newton,d2_xboole_0]), [interesting(0.5),file(newton,e5_2_2_1__newton),[file(newton,e5_2_2_1__newton)]]). fof(e6_2_2_1__newton,plain,( ~ ( ~ r2_hidden(c1_2_2_1__newton,k1_tarski(1)) & ~ r2_hidden(c1_2_2_1__newton,a_1_0_newton(c1_2__newton)) & ~ r2_hidden(c1_2_2_1__newton,k1_tarski(c1_2__newton)) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_2__newton,dt_c1_2_2_1__newton,e4_2_2_1__newton])],[reflexivity_r1_tarski,cc1_finseq_1,fc9_membered,rc1_finseq_1,rc2_finset_1,rc3_finseq_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,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,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_int_1,cc3_membered,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,cc9_membered,fc10_membered,fc11_membered,fc1_ordinal2,fc22_membered,fc23_membered,fc24_membered,fc25_membered,fc26_membered,fc2_finseq_1,fc5_membered,fc6_membered,fc7_membered,fc8_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_boole,t1_real,t2_real,t3_real,t3_subset,t4_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,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_nat_1,cc2_int_1,cc2_nat_1,fc2_membered,fc9_finset_1,rc1_finset_1,rqLessOrEqual__r1_xreal_0__r1_r1,t2_subset,t6_boole,t8_boole,commutativity_k2_xboole_0,idempotence_k2_xboole_0,antisymmetry_r2_hidden,dt_k1_tarski,dt_k2_xboole_0,dt_c1_2__newton,dt_c1_2_2_1__newton,fc1_finset_1,t1_subset,t7_boole,t2_tarski,fraenkel_a_1_0_newton,spc1_numerals,spc1_boole,e5_2_2_1__newton,d2_xboole_0]), [interesting(0.5),file(newton,e6_2_2_1__newton),[file(newton,e6_2_2_1__newton)]]). fof(dh_c1_2_2_1_1__newton,definition, ( ( r2_hidden(c1_2_2_1_1__newton,k1_tarski(1)) => r2_hidden(c1_2_2_1_1__newton,k2_finseq_1(c1_2__newton)) ) => ! [A] : ( r2_hidden(A,k1_tarski(1)) => r2_hidden(A,k2_finseq_1(c1_2__newton)) ) ), introduced(definition,[new_symbol(c1_2_2_1_1__newton),file(newton,c1_2_2_1_1__newton)]), [interesting(0.35),axiom,file(newton,c1_2_2_1_1__newton)]). fof(e1_2_2_1_1__newton,assumption,( r2_hidden(c1_2_2_1_1__newton,k1_tarski(1)) ), introduced(assumption,[file(newton,e1_2_2_1_1__newton)]), [interesting(0.35),axiom,file(newton,e1_2_2_1_1__newton)]). fof(dt_c1_2_2_1_1__newton,assumption,( $true ), introduced(assumption,[file(newton,c1_2_2_1_1__newton)]), [interesting(0.35),axiom,file(newton,c1_2_2_1_1__newton)]). fof(d1_tarski,definition,( ! [A,B] : ( B = k1_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> C = A ) ) ), file(tarski,d1_tarski), [interesting(0.9),axiom,file(tarski,d1_tarski)]). fof(e2_2_2_1_1__newton,plain,( c1_2_2_1_1__newton = 1 ), inference(mizar_by,[status(thm),assumptions([dt_c1_2_2_1_1__newton,e1_2_2_1_1__newton])],[reflexivity_r1_tarski,cc1_finseq_1,fc9_membered,rc1_finseq_1,rc2_finset_1,rc3_finseq_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,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,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_int_1,cc3_membered,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,cc9_membered,fc10_membered,fc11_membered,fc1_ordinal2,fc2_finseq_1,fc5_membered,fc6_membered,fc7_membered,fc8_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t3_subset,t4_subset,t5_subset,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_nat_1,cc2_int_1,cc2_nat_1,fc2_membered,rc1_finset_1,t2_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,dt_k1_tarski,dt_c1_2_2_1_1__newton,fc1_finset_1,t1_subset,t7_boole,spc1_numerals,spc1_boole,e1_2_2_1_1__newton,d1_tarski]), [interesting(0.35),file(newton,e2_2_2_1_1__newton),[file(newton,e2_2_2_1_1__newton)]]). fof(e3_2_2_1_1__newton,plain,( r2_hidden(c1_2_2_1_1__newton,k2_finseq_1(c1_2__newton)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_2__newton,dt_c1_2_2_1_1__newton,e1_2_2_1_1__newton,e1_2__newton])],[cc1_finseq_1,rc1_finseq_1,rc2_finset_1,rc3_finseq_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,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_membered,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_ordinal2,fc2_finseq_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,t2_tarski,fraenkel_a_1_0_finseq_1,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_finseq_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_nat_1,cc1_xreal_0,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_int_1,cc3_nat_1,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_finseq_1,fc2_membered,t1_real,t2_subset,t3_subset,t4_real,t4_subset,t5_subset,t6_boole,t8_boole,d1_finseq_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,redefinition_k2_finseq_1,dt_k2_finseq_1,dt_c1_2__newton,dt_c1_2_2_1_1__newton,t1_subset,t7_boole,spc1_numerals,spc1_boole,e2_2_2_1_1__newton,e1_2__newton,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.35),file(newton,e3_2_2_1_1__newton),[file(newton,e3_2_2_1_1__newton)]]). fof(i3_2_2_1_1__newton,theorem,( $true ), introduced(tautology,[file(newton,i3_2_2_1_1__newton)]), [interesting(0.35),trivial,file(newton,i3_2_2_1_1__newton)]). fof(i2_2_2_1_1__newton,plain,( r2_hidden(c1_2_2_1_1__newton,k2_finseq_1(c1_2__newton)) ), inference(conclusion,[status(thm),assumptions([dt_c1_2__newton,dt_c1_2_2_1_1__newton,e1_2_2_1_1__newton,e1_2__newton])],[e3_2_2_1_1__newton,i3_2_2_1_1__newton]), [interesting(0.35),file(newton,i2_2_2_1_1__newton),[file(newton,i2_2_2_1_1__newton)]]). fof(i1_2_2_1_1__newton,plain, ( r2_hidden(c1_2_2_1_1__newton,k1_tarski(1)) => r2_hidden(c1_2_2_1_1__newton,k2_finseq_1(c1_2__newton)) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_2__newton,dt_c1_2_2_1_1__newton,e1_2__newton]),discharge_asm(discharge,[e1_2_2_1_1__newton])],[e1_2_2_1_1__newton,i2_2_2_1_1__newton]), [interesting(0.35),file(newton,i1_2_2_1_1__newton),[file(newton,i1_2_2_1_1__newton)]]). fof(i1_2_2_1_1_tmp__newton,plain, ( r2_hidden(c1_2_2_1_1__newton,k1_tarski(1)) => r2_hidden(c1_2_2_1_1__newton,k2_finseq_1(c1_2__newton)) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_2__newton,e1_2__newton]),discharge_asm(discharge,[dt_c1_2_2_1_1__newton])],[dt_c1_2_2_1_1__newton,i1_2_2_1_1__newton]), [interesting(0.5),e1_2_2_1__newton]). fof(e1_2_2_1__newton,plain,( ! [A] : ( r2_hidden(A,k1_tarski(1)) => r2_hidden(A,k2_finseq_1(c1_2__newton)) ) ), inference(let,[status(thm),assumptions([dt_c1_2__newton,e1_2__newton])],[i1_2_2_1_1_tmp__newton,dh_c1_2_2_1_1__newton]), [interesting(0.5),file(newton,e1_2_2_1__newton),[file(newton,e1_2_2_1__newton)]]). fof(dh_c1_2_2_1_2__newton,definition, ( ( r2_hidden(c1_2_2_1_2__newton,a_1_0_newton(c1_2__newton)) => r2_hidden(c1_2_2_1_2__newton,k2_finseq_1(c1_2__newton)) ) => ! [A] : ( r2_hidden(A,a_1_0_newton(c1_2__newton)) => r2_hidden(A,k2_finseq_1(c1_2__newton)) ) ), introduced(definition,[new_symbol(c1_2_2_1_2__newton),file(newton,c1_2_2_1_2__newton)]), [interesting(0.35),axiom,file(newton,c1_2_2_1_2__newton)]). fof(e1_2_2_1_2__newton,assumption,( r2_hidden(c1_2_2_1_2__newton,a_1_0_newton(c1_2__newton)) ), introduced(assumption,[file(newton,e1_2_2_1_2__newton)]), [interesting(0.35),axiom,file(newton,e1_2_2_1_2__newton)]). fof(dt_c1_2_2_1_2__newton,assumption,( $true ), introduced(assumption,[file(newton,c1_2_2_1_2__newton)]), [interesting(0.35),axiom,file(newton,c1_2_2_1_2__newton)]). fof(dh_c2_2_2_1_2__newton,definition, ( ? [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) & c1_2_2_1_2__newton = A & ~ r1_xreal_0(A,1) & ~ r1_xreal_0(c1_2__newton,A) ) => ( m2_subset_1(c2_2_2_1_2__newton,k1_numbers,k5_numbers) & c1_2_2_1_2__newton = c2_2_2_1_2__newton & ~ r1_xreal_0(c2_2_2_1_2__newton,1) & ~ r1_xreal_0(c1_2__newton,c2_2_2_1_2__newton) ) ), introduced(definition,[new_symbol(c2_2_2_1_2__newton),file(newton,c2_2_2_1_2__newton)]), [interesting(0.35),axiom,file(newton,c2_2_2_1_2__newton)]). fof(e2_2_2_1_2__newton,plain,( ? [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) & c1_2_2_1_2__newton = A & ~ r1_xreal_0(A,1) & ~ r1_xreal_0(c1_2__newton,A) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_2__newton,dt_c1_2_2_1_2__newton,e1_2_2_1_2__newton])],[cc1_finseq_1,rc1_finseq_1,rc2_finset_1,rc3_finseq_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc2_finseq_1,fc6_membered,rc1_finset_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc5_membered,t1_real,t2_subset,t3_subset,t4_real,t4_subset,t5_subset,t6_boole,t8_boole,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m2_subset_1,dt_c1_2__newton,dt_c1_2_2_1_2__newton,fc2_membered,rqLessOrEqual__r1_xreal_0__r1_r1,t1_subset,t7_boole,t2_tarski,fraenkel_a_1_0_newton,spc1_numerals,spc1_boole,e1_2_2_1_2__newton]), [interesting(0.35),file(newton,e2_2_2_1_2__newton),[file(newton,e2_2_2_1_2__newton)]]). fof(dt_c2_2_2_1_2__newton,plain,( m2_subset_1(c2_2_2_1_2__newton,k1_numbers,k5_numbers) ), inference(consider,[status(thm),assumptions([dt_c1_2__newton,dt_c1_2_2_1_2__newton,e1_2_2_1_2__newton])],[dh_c2_2_2_1_2__newton,e2_2_2_1_2__newton]), [interesting(0.35),file(newton,c2_2_2_1_2__newton),[file(newton,c2_2_2_1_2__newton)]]). fof(e3_2_2_1_2__newton,plain, ( c1_2_2_1_2__newton = c2_2_2_1_2__newton & ~ r1_xreal_0(c2_2_2_1_2__newton,1) & ~ r1_xreal_0(c1_2__newton,c2_2_2_1_2__newton) ), inference(consider,[status(thm),assumptions([dt_c1_2__newton,dt_c1_2_2_1_2__newton,e1_2_2_1_2__newton])],[dh_c2_2_2_1_2__newton,e2_2_2_1_2__newton]), [interesting(0.35),file(newton,e3_2_2_1_2__newton),[file(newton,e3_2_2_1_2__newton)]]). fof(e4_2_2_1_2__newton,plain,( r2_hidden(c1_2_2_1_2__newton,k2_finseq_1(c1_2__newton)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_2__newton,dt_c1_2_2_1_2__newton,e1_2_2_1_2__newton])],[cc1_finseq_1,rc1_finseq_1,rc2_finset_1,rc3_finseq_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,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_membered,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_ordinal2,fc2_finseq_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,t2_tarski,fraenkel_a_1_0_finseq_1,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_finseq_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_nat_1,cc1_xreal_0,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_int_1,cc3_nat_1,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_finseq_1,fc2_membered,t1_real,t2_subset,t3_subset,t4_real,t4_subset,t5_subset,t6_boole,t8_boole,d1_finseq_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,redefinition_k2_finseq_1,dt_k2_finseq_1,dt_c1_2__newton,dt_c1_2_2_1_2__newton,dt_c2_2_2_1_2__newton,t1_subset,t7_boole,spc1_numerals,spc1_boole,e3_2_2_1_2__newton,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.35),file(newton,e4_2_2_1_2__newton),[file(newton,e4_2_2_1_2__newton)]]). fof(i3_2_2_1_2__newton,theorem,( $true ), introduced(tautology,[file(newton,i3_2_2_1_2__newton)]), [interesting(0.35),trivial,file(newton,i3_2_2_1_2__newton)]). fof(i2_2_2_1_2__newton,plain,( r2_hidden(c1_2_2_1_2__newton,k2_finseq_1(c1_2__newton)) ), inference(conclusion,[status(thm),assumptions([dt_c1_2__newton,dt_c1_2_2_1_2__newton,e1_2_2_1_2__newton])],[e4_2_2_1_2__newton,i3_2_2_1_2__newton]), [interesting(0.35),file(newton,i2_2_2_1_2__newton),[file(newton,i2_2_2_1_2__newton)]]). fof(i1_2_2_1_2__newton,plain, ( r2_hidden(c1_2_2_1_2__newton,a_1_0_newton(c1_2__newton)) => r2_hidden(c1_2_2_1_2__newton,k2_finseq_1(c1_2__newton)) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_2__newton,dt_c1_2_2_1_2__newton]),discharge_asm(discharge,[e1_2_2_1_2__newton])],[e1_2_2_1_2__newton,i2_2_2_1_2__newton]), [interesting(0.35),file(newton,i1_2_2_1_2__newton),[file(newton,i1_2_2_1_2__newton)]]). fof(i1_2_2_1_2_tmp__newton,plain, ( r2_hidden(c1_2_2_1_2__newton,a_1_0_newton(c1_2__newton)) => r2_hidden(c1_2_2_1_2__newton,k2_finseq_1(c1_2__newton)) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_2__newton]),discharge_asm(discharge,[dt_c1_2_2_1_2__newton])],[dt_c1_2_2_1_2__newton,i1_2_2_1_2__newton]), [interesting(0.5),e2_2_2_1__newton]). fof(e2_2_2_1__newton,plain,( ! [A] : ( r2_hidden(A,a_1_0_newton(c1_2__newton)) => r2_hidden(A,k2_finseq_1(c1_2__newton)) ) ), inference(let,[status(thm),assumptions([dt_c1_2__newton])],[i1_2_2_1_2_tmp__newton,dh_c1_2_2_1_2__newton]), [interesting(0.5),file(newton,e2_2_2_1__newton),[file(newton,e2_2_2_1__newton)]]). fof(dh_c1_2_2_1_3__newton,definition, ( ( r2_hidden(c1_2_2_1_3__newton,k1_tarski(c1_2__newton)) => r2_hidden(c1_2_2_1_3__newton,k2_finseq_1(c1_2__newton)) ) => ! [A] : ( r2_hidden(A,k1_tarski(c1_2__newton)) => r2_hidden(A,k2_finseq_1(c1_2__newton)) ) ), introduced(definition,[new_symbol(c1_2_2_1_3__newton),file(newton,c1_2_2_1_3__newton)]), [interesting(0.35),axiom,file(newton,c1_2_2_1_3__newton)]). fof(e1_2_2_1_3__newton,assumption,( r2_hidden(c1_2_2_1_3__newton,k1_tarski(c1_2__newton)) ), introduced(assumption,[file(newton,e1_2_2_1_3__newton)]), [interesting(0.35),axiom,file(newton,e1_2_2_1_3__newton)]). 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(t1_numerals,theorem,( m1_subset_1(0,k5_numbers) ), file(numerals,t1_numerals), [interesting(0.9),axiom,file(numerals,t1_numerals)]). fof(dt_c1_2_2_1_3__newton,assumption,( $true ), introduced(assumption,[file(newton,c1_2_2_1_3__newton)]), [interesting(0.35),axiom,file(newton,c1_2_2_1_3__newton)]). 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(spc0_boole,theorem,( v1_xboole_0(0) ), file(boole,spc0_boole), [interesting(0.9),axiom,file(boole,spc0_boole)]). fof(e3_2_2_1_3__newton,plain,( c1_2__newton != 0 ), inference(mizar_by,[status(thm),assumptions([dt_c1_2__newton,e1_2__newton])],[reflexivity_r1_tarski,cc1_finseq_1,rc1_finseq_1,rc2_finset_1,rc3_finseq_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,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc1_ordinal2,fc2_finseq_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_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_nat_1,cc2_int_1,cc2_nat_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,t8_boole,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,dt_c1_2__newton,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,e1_2__newton]), [interesting(0.35),file(newton,e3_2_2_1_3__newton),[file(newton,e3_2_2_1_3__newton)]]). fof(e2_2_2_1_3__newton,plain,( c1_2_2_1_3__newton = c1_2__newton ), inference(mizar_by,[status(thm),assumptions([dt_c1_2__newton,dt_c1_2_2_1_3__newton,e1_2_2_1_3__newton])],[reflexivity_r1_tarski,cc1_finseq_1,cc3_xreal_0,cc6_xreal_0,cc8_xreal_0,fc9_membered,rc1_finseq_1,rc2_finset_1,rc2_xreal_0,rc3_finseq_1,rc3_xreal_0,rc4_xreal_0,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,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,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc7_xreal_0,cc9_membered,fc10_membered,fc11_membered,fc1_ordinal2,fc2_finseq_1,fc5_membered,fc6_membered,fc7_membered,fc8_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc3_finset_1,rc3_nat_1,rc4_finset_1,t3_subset,t4_subset,t5_subset,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_nat_1,cc2_int_1,cc2_nat_1,fc2_membered,rc1_finset_1,t2_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,dt_k1_tarski,dt_c1_2__newton,dt_c1_2_2_1_3__newton,fc1_finset_1,t1_subset,t7_boole,e1_2_2_1_3__newton,d1_tarski]), [interesting(0.35),file(newton,e2_2_2_1_3__newton),[file(newton,e2_2_2_1_3__newton)]]). fof(t5_finseq_1,theorem,( ! [A] : ( v4_ordinal2(A) => ( A = 0 | r2_hidden(A,k2_finseq_1(A)) ) ) ), file(finseq_1,t5_finseq_1), [interesting(0.9),axiom,file(finseq_1,t5_finseq_1)]). fof(e4_2_2_1_3__newton,plain,( r2_hidden(c1_2_2_1_3__newton,k2_finseq_1(c1_2__newton)) ), inference(mizar_by,[status(thm),assumptions([e1_2__newton,dt_c1_2__newton,dt_c1_2_2_1_3__newton,e1_2_2_1_3__newton])],[reflexivity_r1_xreal_0,connectedness_r1_xreal_0,cc1_finseq_1,rc1_finseq_1,rc2_finset_1,rc3_finseq_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,rqLessOrEqual__r1_xreal_0__r1_r1,spc1_boole,spc1_numerals,t1_real,t2_real,t3_real,t4_real,t5_real,t6_real,t7_real,t8_real,spc1_numerals,spc1_boole,reflexivity_r1_tarski,dt_k1_xboole_0,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_membered,cc4_membered,fc1_ordinal2,fc2_finseq_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc2_nat_1,rc3_finset_1,rc3_nat_1,rc4_finset_1,t2_tarski,fraenkel_a_1_0_finseq_1,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_finseq_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,cc9_membered,fc1_finseq_1,fc2_membered,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_numerals,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,d1_finseq_1,antisymmetry_r2_hidden,redefinition_k2_finseq_1,dt_k2_finseq_1,dt_c1_2__newton,dt_c1_2_2_1_3__newton,cc1_xreal_0,cc3_int_1,cc3_nat_1,t1_subset,t7_boole,spc0_numerals,spc0_boole,e3_2_2_1_3__newton,e2_2_2_1_3__newton,t5_finseq_1]), [interesting(0.35),file(newton,e4_2_2_1_3__newton),[file(newton,e4_2_2_1_3__newton)]]). fof(i3_2_2_1_3__newton,theorem,( $true ), introduced(tautology,[file(newton,i3_2_2_1_3__newton)]), [interesting(0.35),trivial,file(newton,i3_2_2_1_3__newton)]). fof(i2_2_2_1_3__newton,plain,( r2_hidden(c1_2_2_1_3__newton,k2_finseq_1(c1_2__newton)) ), inference(conclusion,[status(thm),assumptions([e1_2__newton,dt_c1_2__newton,dt_c1_2_2_1_3__newton,e1_2_2_1_3__newton])],[e4_2_2_1_3__newton,i3_2_2_1_3__newton]), [interesting(0.35),file(newton,i2_2_2_1_3__newton),[file(newton,i2_2_2_1_3__newton)]]). fof(i1_2_2_1_3__newton,plain, ( r2_hidden(c1_2_2_1_3__newton,k1_tarski(c1_2__newton)) => r2_hidden(c1_2_2_1_3__newton,k2_finseq_1(c1_2__newton)) ), inference(discharge_asm,[status(thm),assumptions([e1_2__newton,dt_c1_2__newton,dt_c1_2_2_1_3__newton]),discharge_asm(discharge,[e1_2_2_1_3__newton])],[e1_2_2_1_3__newton,i2_2_2_1_3__newton]), [interesting(0.35),file(newton,i1_2_2_1_3__newton),[file(newton,i1_2_2_1_3__newton)]]). fof(i1_2_2_1_3_tmp__newton,plain, ( r2_hidden(c1_2_2_1_3__newton,k1_tarski(c1_2__newton)) => r2_hidden(c1_2_2_1_3__newton,k2_finseq_1(c1_2__newton)) ), inference(discharge_asm,[status(thm),assumptions([e1_2__newton,dt_c1_2__newton]),discharge_asm(discharge,[dt_c1_2_2_1_3__newton])],[dt_c1_2_2_1_3__newton,i1_2_2_1_3__newton]), [interesting(0.5),e3_2_2_1__newton]). fof(e3_2_2_1__newton,plain,( ! [A] : ( r2_hidden(A,k1_tarski(c1_2__newton)) => r2_hidden(A,k2_finseq_1(c1_2__newton)) ) ), inference(let,[status(thm),assumptions([e1_2__newton,dt_c1_2__newton])],[i1_2_2_1_3_tmp__newton,dh_c1_2_2_1_3__newton]), [interesting(0.5),file(newton,e3_2_2_1__newton),[file(newton,e3_2_2_1__newton)]]). fof(e7_2_2_1__newton,plain,( r2_hidden(c1_2_2_1__newton,k2_finseq_1(c1_2__newton)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_2_2_1__newton,e4_2_2_1__newton,e1_2__newton,dt_c1_2__newton])],[cc1_finseq_1,fc9_membered,rc1_finseq_1,rc2_finset_1,rc3_finseq_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,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_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_membered,fc1_ordinal2,fc2_finseq_1,fc5_membered,fc6_membered,fc7_membered,fc8_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_real,t2_real,t3_real,t4_real,t5_real,t6_real,t7_real,t8_real,fraenkel_a_1_0_finseq_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_finseq_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_nat_1,cc1_xreal_0,cc2_finset_1,cc2_int_1,cc2_nat_1,cc3_int_1,cc3_nat_1,cc6_membered,cc9_membered,fc11_membered,fc1_finseq_1,fc2_membered,rc1_finset_1,rc3_finset_1,rc4_finset_1,rqLessOrEqual__r1_xreal_0__r1_r1,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,d1_finseq_1,antisymmetry_r2_hidden,redefinition_k2_finseq_1,dt_k1_tarski,dt_k2_finseq_1,dt_c1_2__newton,dt_c1_2_2_1__newton,fc1_finset_1,t1_subset,t7_boole,t2_tarski,fraenkel_a_1_0_newton,spc1_numerals,spc1_boole,e6_2_2_1__newton,e1_2_2_1__newton,e2_2_2_1__newton,e3_2_2_1__newton]), [interesting(0.5),file(newton,e7_2_2_1__newton),[file(newton,e7_2_2_1__newton)]]). fof(i3_2_2_1__newton,theorem,( $true ), introduced(tautology,[file(newton,i3_2_2_1__newton)]), [interesting(0.5),trivial,file(newton,i3_2_2_1__newton)]). fof(i2_2_2_1__newton,plain,( r2_hidden(c1_2_2_1__newton,k2_finseq_1(c1_2__newton)) ), inference(conclusion,[status(thm),assumptions([dt_c1_2_2_1__newton,e4_2_2_1__newton,e1_2__newton,dt_c1_2__newton])],[e7_2_2_1__newton,i3_2_2_1__newton]), [interesting(0.5),file(newton,i2_2_2_1__newton),[file(newton,i2_2_2_1__newton)]]). fof(i1_2_2_1__newton,plain, ( r2_hidden(c1_2_2_1__newton,k2_xboole_0(k2_xboole_0(k1_tarski(1),a_1_0_newton(c1_2__newton)),k1_tarski(c1_2__newton))) => r2_hidden(c1_2_2_1__newton,k2_finseq_1(c1_2__newton)) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_2_2_1__newton,e1_2__newton,dt_c1_2__newton]),discharge_asm(discharge,[e4_2_2_1__newton])],[e4_2_2_1__newton,i2_2_2_1__newton]), [interesting(0.5),file(newton,i1_2_2_1__newton),[file(newton,i1_2_2_1__newton)]]). fof(i1_2_2_1_tmp__newton,plain, ( r2_hidden(c1_2_2_1__newton,k2_xboole_0(k2_xboole_0(k1_tarski(1),a_1_0_newton(c1_2__newton)),k1_tarski(c1_2__newton))) => r2_hidden(c1_2_2_1__newton,k2_finseq_1(c1_2__newton)) ), inference(discharge_asm,[status(thm),assumptions([e1_2__newton,dt_c1_2__newton]),discharge_asm(discharge,[dt_c1_2_2_1__newton])],[dt_c1_2_2_1__newton,i1_2_2_1__newton]), [interesting(0.65),e1_2_2__newton]). fof(e1_2_2__newton,plain,( ! [A] : ( r2_hidden(A,k2_xboole_0(k2_xboole_0(k1_tarski(1),a_1_0_newton(c1_2__newton)),k1_tarski(c1_2__newton))) => r2_hidden(A,k2_finseq_1(c1_2__newton)) ) ), inference(let,[status(thm),assumptions([e1_2__newton,dt_c1_2__newton])],[i1_2_2_1_tmp__newton,dh_c1_2_2_1__newton]), [interesting(0.65),file(newton,e1_2_2__newton),[file(newton,e1_2_2__newton)]]). fof(d3_tarski,definition,( ! [A,B] : ( r1_tarski(A,B) <=> ! [C] : ( r2_hidden(C,A) => r2_hidden(C,B) ) ) ), file(tarski,d3_tarski), [interesting(0.9),axiom,file(tarski,d3_tarski)]). fof(e2_2_2__newton,plain,( r1_tarski(k2_xboole_0(k2_xboole_0(k1_tarski(1),a_1_0_newton(c1_2__newton)),k1_tarski(c1_2__newton)),k2_finseq_1(c1_2__newton)) ), inference(mizar_by,[status(thm),assumptions([e1_2__newton,dt_c1_2__newton])],[cc1_finseq_1,fc9_membered,rc1_finseq_1,rc2_finset_1,rc3_finseq_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,dt_k1_xboole_0,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_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_membered,fc1_ordinal2,fc22_membered,fc23_membered,fc24_membered,fc25_membered,fc26_membered,fc2_finseq_1,fc5_membered,fc6_membered,fc7_membered,fc8_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_boole,t1_real,t2_real,t3_real,t4_real,t5_real,t6_real,t7_real,t8_real,fraenkel_a_1_0_finseq_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_finseq_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_nat_1,cc1_xreal_0,cc2_finset_1,cc2_int_1,cc2_nat_1,cc3_int_1,cc3_nat_1,cc6_membered,cc9_membered,fc11_membered,fc1_finseq_1,fc2_membered,fc9_finset_1,rc1_finset_1,rc3_finset_1,rc4_finset_1,rqLessOrEqual__r1_xreal_0__r1_r1,t2_subset,t4_subset,t5_subset,t6_boole,t8_boole,d1_finseq_1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,redefinition_k2_finseq_1,dt_k1_tarski,dt_k2_finseq_1,dt_k2_xboole_0,dt_c1_2__newton,fc1_finset_1,t1_subset,t3_subset,t7_boole,t2_tarski,fraenkel_a_1_0_newton,spc1_numerals,spc1_boole,e1_2_2__newton,d3_tarski]), [interesting(0.65),file(newton,e2_2_2__newton),[file(newton,e2_2_2__newton)]]). fof(i1_2_2__newton,theorem,( $true ), introduced(tautology,[file(newton,i1_2_2__newton)]), [interesting(0.65),trivial,file(newton,i1_2_2__newton)]). fof(e3_2__newton,plain,( r1_tarski(k2_xboole_0(k2_xboole_0(k1_tarski(1),a_1_0_newton(c1_2__newton)),k1_tarski(c1_2__newton)),k2_finseq_1(c1_2__newton)) ), inference(conclusion,[status(thm),assumptions([e1_2__newton,dt_c1_2__newton])],[e2_2_2__newton,i1_2_2__newton]), [interesting(0.8),file(newton,e3_2__newton),[file(newton,e3_2__newton)]]). fof(dt_c1_2_1__newton,assumption,( $true ), introduced(assumption,[file(newton,c1_2_1__newton)]), [interesting(0.65),axiom,file(newton,c1_2_1__newton)]). fof(dh_c1_2_1__newton,definition, ( ~ ( r2_hidden(c1_2_1__newton,k2_finseq_1(c1_2__newton)) & ~ r2_hidden(c1_2_1__newton,k2_xboole_0(k2_xboole_0(k1_tarski(1),a_1_0_newton(c1_2__newton)),k1_tarski(c1_2__newton))) ) => ! [A] : ~ ( r2_hidden(A,k2_finseq_1(c1_2__newton)) & ~ r2_hidden(A,k2_xboole_0(k2_xboole_0(k1_tarski(1),a_1_0_newton(c1_2__newton)),k1_tarski(c1_2__newton))) ) ), introduced(definition,[new_symbol(c1_2_1__newton),file(newton,c1_2_1__newton)]), [interesting(0.65),axiom,file(newton,c1_2_1__newton)]). fof(e1_2_1__newton,assumption,( r2_hidden(c1_2_1__newton,k2_finseq_1(c1_2__newton)) ), introduced(assumption,[file(newton,e1_2_1__newton)]), [interesting(0.65),axiom,file(newton,e1_2_1__newton)]). fof(e1_2_1_1_1__newton,assumption,( ~ r1_xreal_0(c1_2__newton,1) ), introduced(assumption,[file(newton,e1_2_1_1_1__newton)]), [interesting(0.35),axiom,file(newton,e1_2_1_1_1__newton)]). fof(de_c1_2_1_1_1__newton,definition,( c1_2_1_1_1__newton = c1_2_1__newton ), introduced(definition,[new_symbol(c1_2_1_1_1__newton),file(newton,c1_2_1_1_1__newton)]), [interesting(0.35),axiom,file(newton,c1_2_1_1_1__newton)]). fof(e2_2_1_1_1__newton,plain,( m2_subset_1(c1_2_1__newton,k1_numbers,k5_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_2__newton,dt_c1_2_1__newton,e1_2_1__newton])],[reflexivity_r1_xreal_0,connectedness_r1_xreal_0,cc1_finseq_1,cc3_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_finseq_1,rc2_finset_1,rc2_xreal_0,rc3_finseq_1,rc3_xreal_0,rc4_xreal_0,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,rqLessOrEqual__r1_xreal_0__r1_r1,spc1_boole,spc1_numerals,t1_real,t2_real,t3_real,t4_real,t5_real,t6_real,t7_real,t8_real,spc1_numerals,spc1_boole,reflexivity_r1_tarski,dt_k1_xboole_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc7_xreal_0,fc2_finseq_1,fc6_membered,rc1_finset_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc3_finset_1,rc3_nat_1,rc4_finset_1,t2_tarski,fraenkel_a_1_0_finseq_1,existence_m1_subset_1,dt_k1_finseq_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_nat_1,cc1_xreal_0,cc2_int_1,cc2_nat_1,cc3_int_1,cc3_nat_1,cc4_membered,cc6_membered,cc9_membered,fc1_finseq_1,fc1_ordinal2,fc5_membered,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,d1_finseq_1,antisymmetry_r2_hidden,existence_m2_subset_1,redefinition_k2_finseq_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_finseq_1,dt_k5_numbers,dt_m2_subset_1,dt_c1_2__newton,dt_c1_2_1__newton,fc2_membered,t1_subset,t7_boole,e1_2_1__newton]), [interesting(0.35),file(newton,e2_2_1_1_1__newton),[file(newton,e2_2_1_1_1__newton)]]). fof(dt_c1_2_1_1_1__newton,plain,( m2_subset_1(c1_2_1_1_1__newton,k1_numbers,k5_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_2__newton,dt_c1_2_1__newton,e1_2_1__newton])],[cc1_finseq_1,cc3_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_finseq_1,rc2_finset_1,rc2_xreal_0,rc3_finseq_1,rc3_xreal_0,rc4_xreal_0,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc7_xreal_0,fc2_finseq_1,fc6_membered,rc1_finset_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc3_finset_1,rc3_nat_1,rc4_finset_1,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_nat_1,cc2_int_1,cc2_nat_1,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc5_membered,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m2_subset_1,dt_c1_2_1__newton,fc2_membered,de_c1_2_1_1_1__newton,e2_2_1_1_1__newton]), [interesting(0.35),file(newton,c1_2_1_1_1__newton),[file(newton,c1_2_1_1_1__newton)]]). fof(t3_finseq_1,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( r2_hidden(A,k2_finseq_1(B)) <=> ( r1_xreal_0(1,A) & r1_xreal_0(A,B) ) ) ) ) ), file(finseq_1,t3_finseq_1), [interesting(0.9),axiom,file(finseq_1,t3_finseq_1)]). fof(e3_2_1_1_1__newton,plain, ( r1_xreal_0(1,c1_2_1_1_1__newton) & r1_xreal_0(c1_2_1_1_1__newton,c1_2__newton) ), inference(mizar_by,[status(thm),assumptions([dt_c1_2__newton,dt_c1_2_1__newton,e1_2_1__newton])],[cc1_finseq_1,rc1_finseq_1,rc2_finset_1,rc3_finseq_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,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_membered,cc4_membered,fc1_ordinal2,fc2_finseq_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc2_nat_1,rc3_finset_1,rc3_nat_1,rc4_finset_1,t2_tarski,fraenkel_a_1_0_finseq_1,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_finseq_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,cc9_membered,fc1_finseq_1,fc2_membered,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_real,t2_real,t2_subset,t3_real,t3_subset,t4_real,t4_subset,t5_real,t5_subset,t6_boole,t6_real,t7_real,t8_boole,t8_real,d1_finseq_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,redefinition_k2_finseq_1,dt_k2_finseq_1,dt_c1_2__newton,dt_c1_2_1__newton,dt_c1_2_1_1_1__newton,de_c1_2_1_1_1__newton,cc1_xreal_0,cc3_int_1,cc3_nat_1,rqLessOrEqual__r1_xreal_0__r1_r1,t1_subset,t7_boole,spc1_numerals,spc1_boole,e1_2_1__newton,t3_finseq_1]), [interesting(0.35),file(newton,e3_2_1_1_1__newton),[file(newton,e3_2_1_1_1__newton)]]). fof(d5_real_1,definition,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) <=> ~ ( r1_xreal_0(B,A) & B != A ) ) ) ) ), file(real_1,d5_real_1), [interesting(0.9),axiom,file(real_1,d5_real_1)]). fof(e4_2_1_1_1__newton,plain,( ~ ( 1 != c1_2_1_1_1__newton & ~ ( ~ r1_xreal_0(c1_2_1_1_1__newton,1) & ~ r1_xreal_0(c1_2__newton,c1_2_1_1_1__newton) ) & c1_2_1_1_1__newton != c1_2__newton & 1 != c1_2__newton ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_2__newton,dt_c1_2_1__newton,e1_2_1__newton])],[reflexivity_r1_tarski,cc1_finseq_1,rc1_finseq_1,rc2_finset_1,rc3_finseq_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,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc1_ordinal2,fc2_finseq_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_int_1,rc1_membered,rc1_nat_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_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,dt_c1_2_1__newton,cc15_membered,cc1_finset_1,cc1_nat_1,cc2_int_1,cc2_nat_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc2_membered,rc1_xreal_0,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,dt_c1_2__newton,dt_c1_2_1_1_1__newton,de_c1_2_1_1_1__newton,cc2_xreal_0,spc1_numerals,spc1_boole,e3_2_1_1_1__newton,d5_real_1,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.35),file(newton,e4_2_1_1_1__newton),[file(newton,e4_2_1_1_1__newton)]]). fof(e5_2_1_1_1__newton,plain,( ~ ( ~ r2_hidden(c1_2_1__newton,k1_tarski(1)) & ~ r2_hidden(c1_2_1__newton,a_1_0_newton(c1_2__newton)) & ~ r2_hidden(c1_2_1__newton,k1_tarski(c1_2__newton)) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_2__newton,dt_c1_2_1__newton,e1_2_1__newton,e1_2_1_1_1__newton])],[reflexivity_r1_tarski,cc1_finseq_1,fc9_membered,rc1_finseq_1,rc2_finset_1,rc3_finseq_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,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,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_membered,fc11_membered,fc1_ordinal2,fc2_finseq_1,fc5_membered,fc6_membered,fc7_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,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_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc2_membered,fc8_membered,rc1_finset_1,t1_real,t2_subset,t4_real,t6_boole,t8_boole,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,dt_k1_tarski,dt_c1_2__newton,dt_c1_2_1__newton,dt_c1_2_1_1_1__newton,de_c1_2_1_1_1__newton,fc1_finset_1,rqLessOrEqual__r1_xreal_0__r1_r1,t1_subset,t7_boole,t2_tarski,fraenkel_a_1_0_newton,spc1_numerals,spc1_boole,e4_2_1_1_1__newton,e1_2_1_1_1__newton,d1_tarski]), [interesting(0.35),file(newton,e5_2_1_1_1__newton),[file(newton,e5_2_1_1_1__newton)]]). fof(e6_2_1_1_1__newton,plain, ( r2_hidden(c1_2_1__newton,k2_xboole_0(k1_tarski(1),a_1_0_newton(c1_2__newton))) | r2_hidden(c1_2_1__newton,k1_tarski(c1_2__newton)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_2__newton,dt_c1_2_1__newton,e1_2_1__newton,e1_2_1_1_1__newton])],[reflexivity_r1_tarski,cc1_finseq_1,fc9_membered,rc1_finseq_1,rc2_finset_1,rc3_finseq_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,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,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_int_1,cc3_membered,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,cc9_membered,fc10_membered,fc11_membered,fc1_ordinal2,fc22_membered,fc23_membered,fc24_membered,fc25_membered,fc26_membered,fc2_finseq_1,fc5_membered,fc6_membered,fc7_membered,fc8_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_boole,t1_real,t2_real,t3_real,t3_subset,t4_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,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_nat_1,cc2_int_1,cc2_nat_1,fc2_membered,fc9_finset_1,rc1_finset_1,rqLessOrEqual__r1_xreal_0__r1_r1,t2_subset,t6_boole,t8_boole,commutativity_k2_xboole_0,idempotence_k2_xboole_0,antisymmetry_r2_hidden,dt_k1_tarski,dt_k2_xboole_0,dt_c1_2__newton,dt_c1_2_1__newton,fc1_finset_1,t1_subset,t7_boole,t2_tarski,fraenkel_a_1_0_newton,spc1_numerals,spc1_boole,e5_2_1_1_1__newton,d2_xboole_0]), [interesting(0.35),file(newton,e6_2_1_1_1__newton),[file(newton,e6_2_1_1_1__newton)]]). fof(e7_2_1_1_1__newton,plain,( r2_hidden(c1_2_1__newton,k2_xboole_0(k2_xboole_0(k1_tarski(1),a_1_0_newton(c1_2__newton)),k1_tarski(c1_2__newton))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_2__newton,dt_c1_2_1__newton,e1_2_1__newton,e1_2_1_1_1__newton])],[reflexivity_r1_tarski,cc1_finseq_1,fc9_membered,rc1_finseq_1,rc2_finset_1,rc3_finseq_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,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,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_int_1,cc3_membered,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,cc9_membered,fc10_membered,fc11_membered,fc1_ordinal2,fc22_membered,fc23_membered,fc24_membered,fc25_membered,fc26_membered,fc2_finseq_1,fc5_membered,fc6_membered,fc7_membered,fc8_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_boole,t1_real,t2_real,t3_real,t3_subset,t4_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,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_nat_1,cc2_int_1,cc2_nat_1,fc2_membered,fc9_finset_1,rc1_finset_1,rqLessOrEqual__r1_xreal_0__r1_r1,t2_subset,t6_boole,t8_boole,commutativity_k2_xboole_0,idempotence_k2_xboole_0,antisymmetry_r2_hidden,dt_k1_tarski,dt_k2_xboole_0,dt_c1_2__newton,dt_c1_2_1__newton,fc1_finset_1,t1_subset,t7_boole,t2_tarski,fraenkel_a_1_0_newton,spc1_numerals,spc1_boole,e6_2_1_1_1__newton,d2_xboole_0]), [interesting(0.35),file(newton,e7_2_1_1_1__newton),[file(newton,e7_2_1_1_1__newton)]]). fof(i2_2_1_1_1__newton,theorem,( $true ), introduced(tautology,[file(newton,i2_2_1_1_1__newton)]), [interesting(0.35),trivial,file(newton,i2_2_1_1_1__newton)]). fof(i1_2_1_1_1__newton,plain,( r2_hidden(c1_2_1__newton,k2_xboole_0(k2_xboole_0(k1_tarski(1),a_1_0_newton(c1_2__newton)),k1_tarski(c1_2__newton))) ), inference(conclusion,[status(thm),assumptions([dt_c1_2__newton,dt_c1_2_1__newton,e1_2_1__newton,e1_2_1_1_1__newton])],[e7_2_1_1_1__newton,i2_2_1_1_1__newton]), [interesting(0.35),file(newton,i1_2_1_1_1__newton),[file(newton,i1_2_1_1_1__newton)]]). fof(i1_2_1_1__newton,plain, ( ~ r1_xreal_0(c1_2__newton,1) => r2_hidden(c1_2_1__newton,k2_xboole_0(k2_xboole_0(k1_tarski(1),a_1_0_newton(c1_2__newton)),k1_tarski(c1_2__newton))) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_2__newton,dt_c1_2_1__newton,e1_2_1__newton]),discharge_asm(discharge,[e1_2_1_1_1__newton])],[e1_2_1_1_1__newton,i1_2_1_1_1__newton]), [interesting(0.5),file(newton,i1_2_1_1__newton),[file(newton,i1_2_1_1__newton)]]). fof(e1_2_1_1_2__newton,assumption,( c1_2__newton = 1 ), introduced(assumption,[file(newton,e1_2_1_1_2__newton)]), [interesting(0.35),axiom,file(newton,e1_2_1_1_2__newton)]). fof(fc12_membered,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => v1_membered(k2_tarski(A,B)) ) ), file(membered,fc12_membered), [interesting(0.9),axiom,file(membered,fc12_membered)]). fof(fc13_membered,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) ) ) ), file(membered,fc13_membered), [interesting(0.9),axiom,file(membered,fc13_membered)]). fof(fc14_membered,theorem,( ! [A,B] : ( ( v1_rat_1(A) & v1_rat_1(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) & v3_membered(k2_tarski(A,B)) ) ) ), file(membered,fc14_membered), [interesting(0.9),axiom,file(membered,fc14_membered)]). fof(fc15_membered,theorem,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) & v3_membered(k2_tarski(A,B)) & v4_membered(k2_tarski(A,B)) ) ) ), file(membered,fc15_membered), [interesting(0.9),axiom,file(membered,fc15_membered)]). fof(fc16_membered,theorem,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) & v3_membered(k2_tarski(A,B)) & v4_membered(k2_tarski(A,B)) & v5_membered(k2_tarski(A,B)) ) ) ), file(membered,fc16_membered), [interesting(0.9),axiom,file(membered,fc16_membered)]). fof(rqLessOrEqual__r1_xreal_0__r0_r2,theorem,( r1_xreal_0(0,2) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_r2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_r2)]). fof(rqLessOrEqual__r1_xreal_0__r1_r2,theorem,( r1_xreal_0(1,2) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_r2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_r2)]). fof(rqLessOrEqual__r1_xreal_0__r2_r0,theorem,( ~ r1_xreal_0(2,0) ), file(arithm,rqLessOrEqual__r1_xreal_0__r2_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r2_r0)]). fof(rqLessOrEqual__r1_xreal_0__r2_r1,theorem,( ~ r1_xreal_0(2,1) ), file(arithm,rqLessOrEqual__r1_xreal_0__r2_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r2_r1)]). fof(rqLessOrEqual__r1_xreal_0__r2_r2,theorem,( r1_xreal_0(2,2) ), file(arithm,rqLessOrEqual__r1_xreal_0__r2_r2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r2_r2)]). fof(commutativity_k2_tarski,theorem,( ! [A,B] : k2_tarski(A,B) = k2_tarski(B,A) ), file(tarski,k2_tarski), [interesting(0.9),axiom,file(tarski,k2_tarski)]). fof(dt_k2_tarski,axiom,( $true ), file(tarski,k2_tarski), [interesting(0.9),axiom,file(tarski,k2_tarski)]). fof(fc2_finset_1,theorem,( ! [A,B] : ( ~ v1_xboole_0(k2_tarski(A,B)) & v1_finset_1(k2_tarski(A,B)) ) ), file(finset_1,fc2_finset_1), [interesting(0.9),axiom,file(finset_1,fc2_finset_1)]). fof(spc2_numerals,theorem, ( v2_xreal_0(2) & m2_subset_1(2,k1_numbers,k5_numbers) & m1_subset_1(2,k5_numbers) & m1_subset_1(2,k1_numbers) ), file(numerals,spc2_numerals), [interesting(0.9),axiom,file(numerals,spc2_numerals)]). fof(spc2_boole,theorem,( ~ v1_xboole_0(2) ), file(boole,spc2_boole), [interesting(0.9),axiom,file(boole,spc2_boole)]). fof(t4_finseq_1,theorem, ( k2_finseq_1(0) = k1_xboole_0 & k2_finseq_1(1) = k1_tarski(1) & k2_finseq_1(2) = k2_tarski(1,2) ), file(finseq_1,t4_finseq_1), [interesting(0.9),axiom,file(finseq_1,t4_finseq_1)]). fof(e2_2_1_1_2__newton,plain,( r2_hidden(c1_2_1__newton,k2_xboole_0(k2_xboole_0(k1_tarski(1),a_1_0_newton(c1_2__newton)),k1_tarski(c1_2__newton))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_2__newton,dt_c1_2_1__newton,e1_2_1_1_2__newton,e1_2_1__newton])],[reflexivity_r1_tarski,dt_k5_ordinal2,cc2_xreal_0,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_membered,fc12_membered,fc13_membered,fc14_membered,fc15_membered,fc1_ordinal2,fc5_membered,fc7_membered,fc8_membered,fc9_membered,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_finset_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_real,t2_real,t3_real,t4_real,t5_real,t6_real,t7_real,t8_real,fraenkel_a_1_0_finseq_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_finseq_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_finseq_1,cc1_finset_1,cc1_membered,cc1_nat_1,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_int_1,cc2_membered,cc2_nat_1,cc3_int_1,cc3_membered,cc3_nat_1,cc4_membered,cc6_membered,cc9_membered,fc11_membered,fc16_membered,fc1_finseq_1,fc22_membered,fc23_membered,fc24_membered,fc25_membered,fc26_membered,fc2_membered,fc9_finset_1,rc1_finseq_1,rc1_finset_1,rc1_membered,rc3_finseq_1,rc3_finset_1,rc4_finset_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__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,t1_numerals,t2_subset,t3_subset,t4_subset,t5_subset,t8_boole,d1_finseq_1,commutativity_k2_tarski,commutativity_k2_xboole_0,idempotence_k2_xboole_0,antisymmetry_r2_hidden,redefinition_k2_finseq_1,dt_k1_tarski,dt_k1_xboole_0,dt_k2_finseq_1,dt_k2_tarski,dt_k2_xboole_0,dt_c1_2__newton,dt_c1_2_1__newton,fc1_finset_1,fc2_finseq_1,fc2_finset_1,fc6_membered,t1_boole,t1_subset,t6_boole,t7_boole,t2_tarski,fraenkel_a_1_0_newton,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e1_2_1_1_2__newton,e1_2_1__newton,t4_finseq_1,d2_xboole_0]), [interesting(0.35),file(newton,e2_2_1_1_2__newton),[file(newton,e2_2_1_1_2__newton)]]). fof(i2_2_1_1_2__newton,theorem,( $true ), introduced(tautology,[file(newton,i2_2_1_1_2__newton)]), [interesting(0.35),trivial,file(newton,i2_2_1_1_2__newton)]). fof(i1_2_1_1_2__newton,plain,( r2_hidden(c1_2_1__newton,k2_xboole_0(k2_xboole_0(k1_tarski(1),a_1_0_newton(c1_2__newton)),k1_tarski(c1_2__newton))) ), inference(conclusion,[status(thm),assumptions([dt_c1_2__newton,dt_c1_2_1__newton,e1_2_1_1_2__newton,e1_2_1__newton])],[e2_2_1_1_2__newton,i2_2_1_1_2__newton]), [interesting(0.35),file(newton,i1_2_1_1_2__newton),[file(newton,i1_2_1_1_2__newton)]]). fof(i2_2_1_1__newton,plain, ( c1_2__newton = 1 => r2_hidden(c1_2_1__newton,k2_xboole_0(k2_xboole_0(k1_tarski(1),a_1_0_newton(c1_2__newton)),k1_tarski(c1_2__newton))) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_2__newton,dt_c1_2_1__newton,e1_2_1__newton]),discharge_asm(discharge,[e1_2_1_1_2__newton])],[e1_2_1_1_2__newton,i1_2_1_1_2__newton]), [interesting(0.5),file(newton,i2_2_1_1__newton),[file(newton,i2_2_1_1__newton)]]). fof(e1_2_1_1__newton,plain,( ~ ( r1_xreal_0(c1_2__newton,1) & c1_2__newton != 1 ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_2__newton,e1_2__newton])],[reflexivity_r1_tarski,cc1_finseq_1,rc1_finseq_1,rc2_finset_1,rc3_finseq_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,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc1_ordinal2,fc2_finseq_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_int_1,rc1_membered,rc1_nat_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_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_nat_1,cc2_int_1,cc2_nat_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc2_membered,rc1_xreal_0,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,dt_c1_2__newton,cc2_xreal_0,spc1_numerals,spc1_boole,e1_2__newton,d5_real_1,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.5),file(newton,e1_2_1_1__newton),[file(newton,e1_2_1_1__newton)]]). fof(i2_2_1__newton,plain,( r2_hidden(c1_2_1__newton,k2_xboole_0(k2_xboole_0(k1_tarski(1),a_1_0_newton(c1_2__newton)),k1_tarski(c1_2__newton))) ), inference(percases,[status(thm),assumptions([dt_c1_2_1__newton,e1_2_1__newton,dt_c1_2__newton,e1_2__newton])],[i1_2_1_1__newton,i2_2_1_1__newton,e1_2_1_1__newton]), [interesting(0.65),file(newton,i2_2_1__newton),[file(newton,i2_2_1__newton)]]). fof(i1_2_1__newton,plain,( ~ ( r2_hidden(c1_2_1__newton,k2_finseq_1(c1_2__newton)) & ~ r2_hidden(c1_2_1__newton,k2_xboole_0(k2_xboole_0(k1_tarski(1),a_1_0_newton(c1_2__newton)),k1_tarski(c1_2__newton))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_2_1__newton,dt_c1_2__newton,e1_2__newton]),discharge_asm(discharge,[e1_2_1__newton])],[e1_2_1__newton,i2_2_1__newton]), [interesting(0.65),file(newton,i1_2_1__newton),[file(newton,i1_2_1__newton)]]). fof(i1_2_1_tmp__newton,plain,( ~ ( r2_hidden(c1_2_1__newton,k2_finseq_1(c1_2__newton)) & ~ r2_hidden(c1_2_1__newton,k2_xboole_0(k2_xboole_0(k1_tarski(1),a_1_0_newton(c1_2__newton)),k1_tarski(c1_2__newton))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_2__newton,e1_2__newton]),discharge_asm(discharge,[dt_c1_2_1__newton])],[dt_c1_2_1__newton,i1_2_1__newton]), [interesting(0.8),e2_2__newton]). fof(e2_2__newton,plain,( r1_tarski(k2_finseq_1(c1_2__newton),k2_xboole_0(k2_xboole_0(k1_tarski(1),a_1_0_newton(c1_2__newton)),k1_tarski(c1_2__newton))) ), inference(let,[status(thm),assumptions([dt_c1_2__newton,e1_2__newton])],[i1_2_1_tmp__newton,fc9_membered,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_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_membered,fc1_ordinal2,fc22_membered,fc23_membered,fc24_membered,fc25_membered,fc26_membered,fc5_membered,fc7_membered,fc8_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_finseq_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_nat_1,cc1_xreal_0,cc2_finset_1,cc2_int_1,cc2_nat_1,cc3_int_1,cc3_nat_1,cc6_membered,cc9_membered,fc11_membered,fc1_finseq_1,fc2_membered,fc9_finset_1,rc1_finset_1,rc3_finset_1,rc4_finset_1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,redefinition_k2_finseq_1,dt_k1_tarski,dt_k2_finseq_1,dt_k2_xboole_0,dt_c1_2__newton,fc1_finset_1,spc1_numerals,spc1_boole,t2_tarski,fraenkel_a_1_0_newton,d3_tarski,dh_c1_2_1__newton]), [interesting(0.8),file(newton,e2_2__newton),[file(newton,e2_2__newton)]]). fof(d10_xboole_0,definition,( ! [A,B] : ( A = B <=> ( r1_tarski(A,B) & r1_tarski(B,A) ) ) ), file(xboole_0,d10_xboole_0), [interesting(0.9),axiom,file(xboole_0,d10_xboole_0)]). fof(e4_2__newton,plain,( k2_finseq_1(c1_2__newton) = k2_xboole_0(k2_xboole_0(k1_tarski(1),a_1_0_newton(c1_2__newton)),k1_tarski(c1_2__newton)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_2__newton,e1_2__newton])],[cc1_finseq_1,fc9_membered,rc1_finseq_1,rc2_finset_1,rc3_finseq_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,dt_k1_xboole_0,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_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_membered,fc1_ordinal2,fc22_membered,fc23_membered,fc24_membered,fc25_membered,fc26_membered,fc2_finseq_1,fc5_membered,fc6_membered,fc7_membered,fc8_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_boole,t1_real,t2_real,t3_real,t4_real,t5_real,t6_real,t7_real,t8_real,fraenkel_a_1_0_finseq_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_finseq_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_nat_1,cc1_xreal_0,cc2_finset_1,cc2_int_1,cc2_nat_1,cc3_int_1,cc3_nat_1,cc6_membered,cc9_membered,fc11_membered,fc1_finseq_1,fc2_membered,fc9_finset_1,rc1_finset_1,rc3_finset_1,rc4_finset_1,rqLessOrEqual__r1_xreal_0__r1_r1,t1_subset,t2_subset,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,d1_finseq_1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,reflexivity_r1_tarski,redefinition_k2_finseq_1,dt_k1_tarski,dt_k2_finseq_1,dt_k2_xboole_0,dt_c1_2__newton,fc1_finset_1,t3_subset,t2_tarski,fraenkel_a_1_0_newton,spc1_numerals,spc1_boole,e3_2__newton,e2_2__newton,d10_xboole_0]), [interesting(0.8),file(newton,e4_2__newton),[file(newton,e4_2__newton)]]). fof(i3_2__newton,theorem,( $true ), introduced(tautology,[file(newton,i3_2__newton)]), [interesting(0.8),trivial,file(newton,i3_2__newton)]). fof(i2_2__newton,plain,( k2_finseq_1(c1_2__newton) = k2_xboole_0(k2_xboole_0(k1_tarski(1),a_1_0_newton(c1_2__newton)),k1_tarski(c1_2__newton)) ), inference(conclusion,[status(thm),assumptions([dt_c1_2__newton,e1_2__newton])],[e4_2__newton,i3_2__newton]), [interesting(0.8),file(newton,i2_2__newton),[file(newton,i2_2__newton)]]). fof(i1_2__newton,plain, ( r1_xreal_0(1,c1_2__newton) => k2_finseq_1(c1_2__newton) = k2_xboole_0(k2_xboole_0(k1_tarski(1),a_1_0_newton(c1_2__newton)),k1_tarski(c1_2__newton)) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_2__newton]),discharge_asm(discharge,[e1_2__newton])],[e1_2__newton,i2_2__newton]), [interesting(0.8),file(newton,i1_2__newton),[file(newton,i1_2__newton)]]). fof(i1_2_tmp__newton,plain, ( m2_subset_1(c1_2__newton,k1_numbers,k5_numbers) => ( r1_xreal_0(1,c1_2__newton) => k2_finseq_1(c1_2__newton) = k2_xboole_0(k2_xboole_0(k1_tarski(1),a_1_0_newton(c1_2__newton)),k1_tarski(c1_2__newton)) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_2__newton])],[dt_c1_2__newton,i1_2__newton]), [interesting(1),t5_newton]). fof(t5_newton,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r1_xreal_0(1,A) => k2_finseq_1(A) = k2_xboole_0(k2_xboole_0(k1_tarski(1),a_1_0_newton(A)),k1_tarski(A)) ) ) ), inference(let,[status(thm),assumptions([])],[i1_2_tmp__newton,dh_c1_2__newton]), [interesting(1),file(newton,t5_newton),[file(newton,t5_newton)]]).