% Mizar ND problem: t51_qc_lang1,qc_lang1,1227,57 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(dt_m1_relset_1,axiom,( $true ), file(relset_1,m1_relset_1), [interesting(0.9),axiom,file(relset_1,m1_relset_1)]). 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(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(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(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(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(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(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(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(reflexivity_r1_tarski,theorem,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), [interesting(0.9),axiom,file(tarski,r1_tarski)]). 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_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(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_k1_tarski,axiom,( $true ), file(tarski,k1_tarski), [interesting(0.9),axiom,file(tarski,k1_tarski)]). 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_k2_tarski,axiom,( $true ), file(tarski,k2_tarski), [interesting(0.9),axiom,file(tarski,k2_tarski)]). 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_k5_finseq_1,axiom,( $true ), file(finseq_1,k5_finseq_1), [interesting(0.9),axiom,file(finseq_1,k5_finseq_1)]). fof(dt_k6_qc_lang1,axiom,( ! [A] : ( m1_subset_1(A,k5_qc_lang1) => m2_subset_1(k6_qc_lang1(A),k1_numbers,k5_numbers) ) ), file(qc_lang1,k6_qc_lang1), [interesting(0.9),axiom,file(qc_lang1,k6_qc_lang1)]). 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(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(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(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_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_membered,theorem,( ! [A] : ( v5_membered(A) => v4_membered(A) ) ), file(membered,cc1_membered), [interesting(0.9),axiom,file(membered,cc1_membered)]). 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(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_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_membered,theorem,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), [interesting(0.9),axiom,file(membered,cc2_membered)]). fof(cc3_membered,theorem,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), [interesting(0.9),axiom,file(membered,cc3_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(fc12_finseq_1,theorem,( ! [A] : ( ~ v1_xboole_0(k5_finseq_1(A)) & v1_relat_1(k5_finseq_1(A)) & v1_funct_1(k5_finseq_1(A)) & v1_finset_1(k5_finseq_1(A)) & v1_finseq_1(k5_finseq_1(A)) ) ), file(finseq_1,fc12_finseq_1), [interesting(0.9),axiom,file(finseq_1,fc12_finseq_1)]). 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(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(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(fc2_subset_1,theorem,( ! [A] : ~ v1_xboole_0(k1_tarski(A)) ), file(subset_1,fc2_subset_1), [interesting(0.9),axiom,file(subset_1,fc2_subset_1)]). fof(fc3_finseq_1,theorem,( ! [A] : ( v1_relat_1(k5_finseq_1(A)) & v1_funct_1(k5_finseq_1(A)) ) ), file(finseq_1,fc3_finseq_1), [interesting(0.9),axiom,file(finseq_1,fc3_finseq_1)]). fof(fc3_subset_1,theorem,( ! [A,B] : ~ v1_xboole_0(k2_tarski(A,B)) ), file(subset_1,fc3_subset_1), [interesting(0.9),axiom,file(subset_1,fc3_subset_1)]). fof(fc4_finseq_1,theorem,( ! [A] : ( v1_relat_1(k5_finseq_1(A)) & v1_funct_1(k5_finseq_1(A)) & v1_finset_1(k5_finseq_1(A)) & v1_finseq_1(k5_finseq_1(A)) ) ), file(finseq_1,fc4_finseq_1), [interesting(0.9),axiom,file(finseq_1,fc4_finseq_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(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_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_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(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__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__r0_r3,theorem,( r1_xreal_0(0,3) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_r3), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_r3)]). fof(rqLessOrEqual__r1_xreal_0__r0_r7,theorem,( r1_xreal_0(0,7) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_r7), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_r7)]). 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(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__r1_r3,theorem,( r1_xreal_0(1,3) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_r3), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_r3)]). fof(rqLessOrEqual__r1_xreal_0__r1_r7,theorem,( r1_xreal_0(1,7) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_r7), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_r7)]). 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(rqLessOrEqual__r1_xreal_0__r2_r3,theorem,( r1_xreal_0(2,3) ), file(arithm,rqLessOrEqual__r1_xreal_0__r2_r3), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r2_r3)]). fof(rqLessOrEqual__r1_xreal_0__r2_r7,theorem,( r1_xreal_0(2,7) ), file(arithm,rqLessOrEqual__r1_xreal_0__r2_r7), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r2_r7)]). fof(rqLessOrEqual__r1_xreal_0__r3_r0,theorem,( ~ r1_xreal_0(3,0) ), file(arithm,rqLessOrEqual__r1_xreal_0__r3_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r3_r0)]). fof(rqLessOrEqual__r1_xreal_0__r3_r1,theorem,( ~ r1_xreal_0(3,1) ), file(arithm,rqLessOrEqual__r1_xreal_0__r3_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r3_r1)]). fof(rqLessOrEqual__r1_xreal_0__r3_r2,theorem,( ~ r1_xreal_0(3,2) ), file(arithm,rqLessOrEqual__r1_xreal_0__r3_r2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r3_r2)]). fof(rqLessOrEqual__r1_xreal_0__r3_r3,theorem,( r1_xreal_0(3,3) ), file(arithm,rqLessOrEqual__r1_xreal_0__r3_r3), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r3_r3)]). fof(rqLessOrEqual__r1_xreal_0__r3_r7,theorem,( r1_xreal_0(3,7) ), file(arithm,rqLessOrEqual__r1_xreal_0__r3_r7), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r3_r7)]). fof(rqLessOrEqual__r1_xreal_0__r7_r0,theorem,( ~ r1_xreal_0(7,0) ), file(arithm,rqLessOrEqual__r1_xreal_0__r7_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r7_r0)]). fof(rqLessOrEqual__r1_xreal_0__r7_r1,theorem,( ~ r1_xreal_0(7,1) ), file(arithm,rqLessOrEqual__r1_xreal_0__r7_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r7_r1)]). fof(rqLessOrEqual__r1_xreal_0__r7_r2,theorem,( ~ r1_xreal_0(7,2) ), file(arithm,rqLessOrEqual__r1_xreal_0__r7_r2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r7_r2)]). fof(rqLessOrEqual__r1_xreal_0__r7_r3,theorem,( ~ r1_xreal_0(7,3) ), file(arithm,rqLessOrEqual__r1_xreal_0__r7_r3), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r7_r3)]). fof(rqLessOrEqual__r1_xreal_0__r7_r7,theorem,( r1_xreal_0(7,7) ), file(arithm,rqLessOrEqual__r1_xreal_0__r7_r7), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r7_r7)]). fof(spc7_boole,theorem,( ~ v1_xboole_0(7) ), file(boole,spc7_boole), [interesting(0.9),axiom,file(boole,spc7_boole)]). fof(spc7_numerals,theorem, ( v2_xreal_0(7) & m2_subset_1(7,k1_numbers,k5_numbers) & m1_subset_1(7,k5_numbers) & m1_subset_1(7,k1_numbers) ), file(numerals,spc7_numerals), [interesting(0.9),axiom,file(numerals,spc7_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(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(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(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(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_k12_mcart_1,definition,( ! [A,B,C,D] : ( ( m1_subset_1(C,k1_zfmisc_1(A)) & m1_subset_1(D,k1_zfmisc_1(B)) ) => k12_mcart_1(A,B,C,D) = k2_zfmisc_1(C,D) ) ), file(mcart_1,k12_mcart_1), [interesting(0.9),axiom,file(mcart_1,k12_mcart_1)]). fof(redefinition_k9_finseq_1,definition,( ! [A] : k9_finseq_1(A) = k5_finseq_1(A) ), file(finseq_1,k9_finseq_1), [interesting(0.9),axiom,file(finseq_1,k9_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_k12_mcart_1,axiom,( ! [A,B,C,D] : ( ( m1_subset_1(C,k1_zfmisc_1(A)) & m1_subset_1(D,k1_zfmisc_1(B)) ) => m1_subset_1(k12_mcart_1(A,B,C,D),k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ), file(mcart_1,k12_mcart_1), [interesting(0.9),axiom,file(mcart_1,k12_mcart_1)]). 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_k4_tarski,axiom,( $true ), file(tarski,k4_tarski), [interesting(0.9),axiom,file(tarski,k4_tarski)]). fof(dt_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), [interesting(0.9),axiom,file(ordinal2,k5_ordinal2)]). fof(dt_k9_finseq_1,axiom,( ! [A] : ( v1_relat_1(k9_finseq_1(A)) & v1_funct_1(k9_finseq_1(A)) ) ), file(finseq_1,k9_finseq_1), [interesting(0.9),axiom,file(finseq_1,k9_finseq_1)]). 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(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(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(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(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(cc4_membered,theorem,( ! [A] : ( v2_membered(A) => v1_membered(A) ) ), file(membered,cc4_membered), [interesting(0.9),axiom,file(membered,cc4_membered)]). 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(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(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(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(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(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(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(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(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(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_0_0_qc_lang1,definition,( ! [A] : ( r2_hidden(A,a_0_0_qc_lang1) <=> ? [B,C] : ( m2_subset_1(B,k1_numbers,k5_numbers) & m2_subset_1(C,k1_numbers,k5_numbers) & A = k4_tarski(B,C) & r1_xreal_0(7,B) ) ) ), file(qc_lang1,a_0_0_qc_lang1), [interesting(0.9),axiom,file(qc_lang1,a_0_0_qc_lang1)]). fof(fraenkel_a_1_0_qc_lang1,definition,( ! [A,B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( r2_hidden(A,a_1_0_qc_lang1(B)) <=> ? [C] : ( m1_subset_1(C,k5_qc_lang1) & A = C & k6_qc_lang1(C) = B ) ) ) ), file(qc_lang1,a_1_0_qc_lang1), [interesting(0.9),axiom,file(qc_lang1,a_1_0_qc_lang1)]). fof(d5_tarski,definition,( ! [A,B] : k4_tarski(A,B) = k2_tarski(k2_tarski(A,B),k1_tarski(A)) ), file(tarski,d5_tarski), [interesting(0.9),axiom,file(tarski,d5_tarski)]). 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_k10_qc_lang1,axiom,( ! [A] : ( m1_subset_1(A,k8_qc_lang1) => m2_finseq_1(k10_qc_lang1(A),k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers)) ) ), file(qc_lang1,k10_qc_lang1), [interesting(0.9),axiom,file(qc_lang1,k10_qc_lang1)]). fof(dt_k11_qc_lang1,axiom,( m1_subset_1(k11_qc_lang1,k8_qc_lang1) ), file(qc_lang1,k11_qc_lang1), [interesting(0.9),axiom,file(qc_lang1,k11_qc_lang1)]). fof(dt_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), [interesting(0.9),axiom,file(funct_1,k1_funct_1)]). fof(dt_k1_mcart_1,axiom,( $true ), file(mcart_1,k1_mcart_1), [interesting(0.9),axiom,file(mcart_1,k1_mcart_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_k5_qc_lang1,axiom,( $true ), file(qc_lang1,k5_qc_lang1), [interesting(0.9),axiom,file(qc_lang1,k5_qc_lang1)]). fof(dt_k7_qc_lang1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => m1_subset_1(k7_qc_lang1(A),k1_zfmisc_1(k5_qc_lang1)) ) ), file(qc_lang1,k7_qc_lang1), [interesting(0.9),axiom,file(qc_lang1,k7_qc_lang1)]). fof(dt_k8_qc_lang1,axiom,( ~ v1_xboole_0(k8_qc_lang1) ), file(qc_lang1,k8_qc_lang1), [interesting(0.9),axiom,file(qc_lang1,k8_qc_lang1)]). 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(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(fc5_qc_lang1,theorem,( ~ v1_xboole_0(k5_qc_lang1) ), file(qc_lang1,fc5_qc_lang1), [interesting(0.9),axiom,file(qc_lang1,fc5_qc_lang1)]). fof(fc6_qc_lang1,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ~ v1_xboole_0(k7_qc_lang1(A)) ) ), file(qc_lang1,fc6_qc_lang1), [interesting(0.9),axiom,file(qc_lang1,fc6_qc_lang1)]). fof(t1_numerals,theorem,( m1_subset_1(0,k5_numbers) ), file(numerals,t1_numerals), [interesting(0.9),axiom,file(numerals,t1_numerals)]). fof(d12_qc_lang1,definition,( ! [A] : ( m1_subset_1(A,k8_qc_lang1) => k10_qc_lang1(A) = A ) ), file(qc_lang1,d12_qc_lang1), [interesting(0.9),axiom,file(qc_lang1,d12_qc_lang1)]). fof(d13_qc_lang1,definition,( k11_qc_lang1 = k9_finseq_1(k4_tarski(0,0)) ), file(qc_lang1,d13_qc_lang1), [interesting(0.9),axiom,file(qc_lang1,d13_qc_lang1)]). fof(d5_qc_lang1,definition,( k5_qc_lang1 = a_0_0_qc_lang1 ), file(qc_lang1,d5_qc_lang1), [interesting(0.9),axiom,file(qc_lang1,d5_qc_lang1)]). fof(d7_qc_lang1,definition,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k7_qc_lang1(A) = a_1_0_qc_lang1(A) ) ), file(qc_lang1,d7_qc_lang1), [interesting(0.9),axiom,file(qc_lang1,d7_qc_lang1)]). 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(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(spc3_numerals,theorem, ( v2_xreal_0(3) & m2_subset_1(3,k1_numbers,k5_numbers) & m1_subset_1(3,k5_numbers) & m1_subset_1(3,k1_numbers) ), file(numerals,spc3_numerals), [interesting(0.9),axiom,file(numerals,spc3_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(spc2_boole,theorem,( ~ v1_xboole_0(2) ), file(boole,spc2_boole), [interesting(0.9),axiom,file(boole,spc2_boole)]). fof(spc3_boole,theorem,( ~ v1_xboole_0(3) ), file(boole,spc3_boole), [interesting(0.9),axiom,file(boole,spc3_boole)]). fof(t49_qc_lang1,theorem,( ! [A] : ( m1_subset_1(A,k8_qc_lang1) => ( k1_mcart_1(k1_funct_1(k10_qc_lang1(k11_qc_lang1),1)) = 0 & ~ ( v2_qc_lang1(A) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ m2_subset_1(k1_funct_1(k10_qc_lang1(A),1),k5_qc_lang1,k7_qc_lang1(B)) ) ) & ( v3_qc_lang1(A) => k1_mcart_1(k1_funct_1(k10_qc_lang1(A),1)) = 1 ) & ( v4_qc_lang1(A) => k1_mcart_1(k1_funct_1(k10_qc_lang1(A),1)) = 2 ) & ( v5_qc_lang1(A) => k1_mcart_1(k1_funct_1(k10_qc_lang1(A),1)) = 3 ) ) ) ), file(qc_lang1,t49_qc_lang1), [interesting(0.9),axiom,file(qc_lang1,t49_qc_lang1)]). fof(t50_qc_lang1,theorem,( ! [A] : ( m1_subset_1(A,k8_qc_lang1) => ( v2_qc_lang1(A) => ( k1_mcart_1(k1_funct_1(k10_qc_lang1(A),1)) != 0 & k1_mcart_1(k1_funct_1(k10_qc_lang1(A),1)) != 1 & k1_mcart_1(k1_funct_1(k10_qc_lang1(A),1)) != 2 & k1_mcart_1(k1_funct_1(k10_qc_lang1(A),1)) != 3 ) ) ) ), file(qc_lang1,t50_qc_lang1), [interesting(0.9),axiom,file(qc_lang1,t50_qc_lang1)]). fof(e1_45__qc_lang1,plain, ( ~ v2_qc_lang1(k11_qc_lang1) & ~ v3_qc_lang1(k11_qc_lang1) & ~ v4_qc_lang1(k11_qc_lang1) & ~ v5_qc_lang1(k11_qc_lang1) ), inference(mizar_by,[status(thm),assumptions([])],[existence_m1_relset_1,dt_m1_relset_1,fc10_membered,fc14_membered,fc15_membered,fc9_membered,rc2_finseq_1,rc3_finseq_1,rc4_finseq_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,commutativity_k2_tarski,reflexivity_r1_tarski,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_tarski,dt_k1_xboole_0,dt_k2_tarski,dt_k2_zfmisc_1,dt_k5_finseq_1,dt_k6_qc_lang1,dt_m1_finseq_1,dt_m2_relset_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_finseq_1,cc1_membered,cc1_relset_1,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,fc11_membered,fc12_finseq_1,fc12_membered,fc13_membered,fc16_membered,fc2_finseq_1,fc2_subset_1,fc3_finseq_1,fc3_subset_1,fc4_finseq_1,fc4_subset_1,fc6_membered,fc7_membered,fc8_membered,rc1_arytm_3,rc1_finseq_1,rc1_membered,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r0_r3,rqLessOrEqual__r1_xreal_0__r0_r7,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_r3,rqLessOrEqual__r1_xreal_0__r1_r7,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r2_r3,rqLessOrEqual__r1_xreal_0__r2_r7,rqLessOrEqual__r1_xreal_0__r3_r0,rqLessOrEqual__r1_xreal_0__r3_r1,rqLessOrEqual__r1_xreal_0__r3_r2,rqLessOrEqual__r1_xreal_0__r3_r3,rqLessOrEqual__r1_xreal_0__r3_r7,rqLessOrEqual__r1_xreal_0__r7_r0,rqLessOrEqual__r1_xreal_0__r7_r1,rqLessOrEqual__r1_xreal_0__r7_r2,rqLessOrEqual__r1_xreal_0__r7_r3,rqLessOrEqual__r1_xreal_0__r7_r7,spc7_boole,spc7_numerals,t1_real,t1_subset,t4_real,t4_subset,t5_subset,spc7_numerals,spc7_boole,existence_m2_finseq_1,redefinition_k12_mcart_1,redefinition_k9_finseq_1,redefinition_m2_finseq_1,dt_k12_mcart_1,dt_k1_zfmisc_1,dt_k4_tarski,dt_k5_ordinal2,dt_k9_finseq_1,dt_m2_finseq_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc3_arytm_3,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc1_subset_1,fc5_membered,rc1_subset_1,rc2_subset_1,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,t2_tarski,fraenkel_a_0_0_qc_lang1,fraenkel_a_1_0_qc_lang1,d5_tarski,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_qc_lang1,dt_k11_qc_lang1,dt_k1_funct_1,dt_k1_mcart_1,dt_k1_numbers,dt_k5_numbers,dt_k5_qc_lang1,dt_k7_qc_lang1,dt_k8_qc_lang1,dt_m1_subset_1,dt_m2_subset_1,fc2_membered,fc5_qc_lang1,fc6_qc_lang1,t1_numerals,d12_qc_lang1,d13_qc_lang1,d5_qc_lang1,d7_qc_lang1,spc0_numerals,spc1_numerals,spc2_numerals,spc3_numerals,spc0_boole,spc1_boole,spc2_boole,spc3_boole,t49_qc_lang1,t50_qc_lang1]), [interesting(0.8),file(qc_lang1,e1_45__qc_lang1),[file(qc_lang1,e1_45__qc_lang1)]]). fof(dh_c1_45__qc_lang1,definition, ( ( m1_subset_1(c1_45__qc_lang1,k8_qc_lang1) => ( ~ ( v2_qc_lang1(c1_45__qc_lang1) & v3_qc_lang1(c1_45__qc_lang1) ) & ~ ( v2_qc_lang1(c1_45__qc_lang1) & v4_qc_lang1(c1_45__qc_lang1) ) & ~ ( v2_qc_lang1(c1_45__qc_lang1) & v5_qc_lang1(c1_45__qc_lang1) ) & ~ ( v3_qc_lang1(c1_45__qc_lang1) & v4_qc_lang1(c1_45__qc_lang1) ) & ~ ( v3_qc_lang1(c1_45__qc_lang1) & v5_qc_lang1(c1_45__qc_lang1) ) & ~ ( v4_qc_lang1(c1_45__qc_lang1) & v5_qc_lang1(c1_45__qc_lang1) ) ) ) => ! [A] : ( m1_subset_1(A,k8_qc_lang1) => ( ~ ( v2_qc_lang1(A) & v3_qc_lang1(A) ) & ~ ( v2_qc_lang1(A) & v4_qc_lang1(A) ) & ~ ( v2_qc_lang1(A) & v5_qc_lang1(A) ) & ~ ( v3_qc_lang1(A) & v4_qc_lang1(A) ) & ~ ( v3_qc_lang1(A) & v5_qc_lang1(A) ) & ~ ( v4_qc_lang1(A) & v5_qc_lang1(A) ) ) ) ), introduced(definition,[new_symbol(c1_45__qc_lang1),file(qc_lang1,c1_45__qc_lang1)]), [interesting(0.8),axiom,file(qc_lang1,c1_45__qc_lang1)]). fof(dt_c1_45__qc_lang1,assumption,( m1_subset_1(c1_45__qc_lang1,k8_qc_lang1) ), introduced(assumption,[file(qc_lang1,c1_45__qc_lang1)]), [interesting(0.8),axiom,file(qc_lang1,c1_45__qc_lang1)]). fof(e2_45__qc_lang1,plain, ( k1_mcart_1(k1_funct_1(k10_qc_lang1(k11_qc_lang1),1)) = 0 & ( v3_qc_lang1(c1_45__qc_lang1) => k1_mcart_1(k1_funct_1(k10_qc_lang1(c1_45__qc_lang1),1)) = 1 ) & ( v4_qc_lang1(c1_45__qc_lang1) => k1_mcart_1(k1_funct_1(k10_qc_lang1(c1_45__qc_lang1),1)) = 2 ) & ( v5_qc_lang1(c1_45__qc_lang1) => k1_mcart_1(k1_funct_1(k10_qc_lang1(c1_45__qc_lang1),1)) = 3 ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_45__qc_lang1])],[existence_m1_relset_1,dt_m1_relset_1,fc10_membered,fc14_membered,fc15_membered,fc9_membered,rc2_finseq_1,rc3_finseq_1,rc4_finseq_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,commutativity_k2_tarski,reflexivity_r1_tarski,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_tarski,dt_k1_xboole_0,dt_k2_tarski,dt_k2_zfmisc_1,dt_k5_finseq_1,dt_k6_qc_lang1,dt_m1_finseq_1,dt_m2_relset_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_finseq_1,cc1_membered,cc1_relset_1,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,fc11_membered,fc12_finseq_1,fc12_membered,fc13_membered,fc16_membered,fc2_finseq_1,fc2_subset_1,fc3_finseq_1,fc3_subset_1,fc4_finseq_1,fc4_subset_1,fc6_membered,fc7_membered,fc8_membered,rc1_arytm_3,rc1_finseq_1,rc1_membered,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r0_r3,rqLessOrEqual__r1_xreal_0__r0_r7,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_r3,rqLessOrEqual__r1_xreal_0__r1_r7,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r2_r3,rqLessOrEqual__r1_xreal_0__r2_r7,rqLessOrEqual__r1_xreal_0__r3_r0,rqLessOrEqual__r1_xreal_0__r3_r1,rqLessOrEqual__r1_xreal_0__r3_r2,rqLessOrEqual__r1_xreal_0__r3_r3,rqLessOrEqual__r1_xreal_0__r3_r7,rqLessOrEqual__r1_xreal_0__r7_r0,rqLessOrEqual__r1_xreal_0__r7_r1,rqLessOrEqual__r1_xreal_0__r7_r2,rqLessOrEqual__r1_xreal_0__r7_r3,rqLessOrEqual__r1_xreal_0__r7_r7,spc7_boole,spc7_numerals,t1_real,t1_subset,t4_real,t4_subset,t5_subset,spc7_numerals,spc7_boole,existence_m2_finseq_1,redefinition_k12_mcart_1,redefinition_k9_finseq_1,redefinition_m2_finseq_1,dt_k12_mcart_1,dt_k1_zfmisc_1,dt_k4_tarski,dt_k5_ordinal2,dt_k9_finseq_1,dt_m2_finseq_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc3_arytm_3,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc1_subset_1,fc5_membered,rc1_subset_1,rc2_subset_1,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,t2_tarski,fraenkel_a_0_0_qc_lang1,fraenkel_a_1_0_qc_lang1,d5_tarski,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_qc_lang1,dt_k11_qc_lang1,dt_k1_funct_1,dt_k1_mcart_1,dt_k1_numbers,dt_k5_numbers,dt_k5_qc_lang1,dt_k7_qc_lang1,dt_k8_qc_lang1,dt_m1_subset_1,dt_m2_subset_1,dt_c1_45__qc_lang1,fc2_membered,fc5_qc_lang1,fc6_qc_lang1,t1_numerals,d12_qc_lang1,d13_qc_lang1,d5_qc_lang1,d7_qc_lang1,spc0_numerals,spc1_numerals,spc2_numerals,spc3_numerals,spc0_boole,spc1_boole,spc2_boole,spc3_boole,t49_qc_lang1]), [interesting(0.8),file(qc_lang1,e2_45__qc_lang1),[file(qc_lang1,e2_45__qc_lang1)]]). fof(e3_45__qc_lang1,plain, ( ~ ( v2_qc_lang1(c1_45__qc_lang1) & v3_qc_lang1(c1_45__qc_lang1) ) & ~ ( v2_qc_lang1(c1_45__qc_lang1) & v4_qc_lang1(c1_45__qc_lang1) ) & ~ ( v2_qc_lang1(c1_45__qc_lang1) & v5_qc_lang1(c1_45__qc_lang1) ) & ~ ( v3_qc_lang1(c1_45__qc_lang1) & v4_qc_lang1(c1_45__qc_lang1) ) & ~ ( v3_qc_lang1(c1_45__qc_lang1) & v5_qc_lang1(c1_45__qc_lang1) ) & ~ ( v4_qc_lang1(c1_45__qc_lang1) & v5_qc_lang1(c1_45__qc_lang1) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_45__qc_lang1])],[reflexivity_r1_tarski,existence_m1_relset_1,dt_m1_relset_1,cc1_arytm_3,cc2_arytm_3,fc10_membered,fc11_membered,fc12_membered,fc13_membered,fc14_membered,fc15_membered,fc16_membered,fc7_membered,fc8_membered,fc9_membered,rc1_arytm_3,rc2_finseq_1,rc3_finseq_1,rc4_finseq_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,commutativity_k2_tarski,antisymmetry_r2_hidden,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_tarski,dt_k2_zfmisc_1,dt_k5_finseq_1,dt_k5_ordinal2,dt_m1_finseq_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_finseq_1,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc3_arytm_3,cc3_membered,cc4_membered,cc6_membered,cc9_membered,fc12_finseq_1,fc1_ordinal2,fc1_subset_1,fc2_finseq_1,fc2_subset_1,fc3_finseq_1,fc3_subset_1,fc4_finseq_1,fc4_subset_1,fc5_membered,fc6_membered,rc1_finseq_1,rc1_membered,rc1_subset_1,rc2_subset_1,t1_subset,t3_subset,t4_subset,t5_subset,existence_m2_finseq_1,existence_m2_subset_1,redefinition_k12_mcart_1,redefinition_k5_numbers,redefinition_k9_finseq_1,redefinition_m2_finseq_1,redefinition_m2_subset_1,dt_k12_mcart_1,dt_k1_numbers,dt_k4_tarski,dt_k5_numbers,dt_k9_finseq_1,dt_m2_finseq_1,dt_m2_subset_1,cc15_membered,fc2_membered,t1_numerals,t2_subset,t6_boole,t7_boole,t8_boole,d5_tarski,existence_m1_subset_1,dt_k10_qc_lang1,dt_k11_qc_lang1,dt_k1_funct_1,dt_k1_mcart_1,dt_k8_qc_lang1,dt_m1_subset_1,dt_c1_45__qc_lang1,d12_qc_lang1,d13_qc_lang1,spc0_numerals,spc1_numerals,spc2_numerals,spc3_numerals,spc0_boole,spc1_boole,spc2_boole,spc3_boole,e2_45__qc_lang1,t50_qc_lang1]), [interesting(0.8),file(qc_lang1,e3_45__qc_lang1),[file(qc_lang1,e3_45__qc_lang1)]]). fof(i3_45__qc_lang1,theorem,( $true ), introduced(tautology,[file(qc_lang1,i3_45__qc_lang1)]), [interesting(0.8),trivial,file(qc_lang1,i3_45__qc_lang1)]). fof(i2_45__qc_lang1,plain, ( ~ ( v2_qc_lang1(c1_45__qc_lang1) & v3_qc_lang1(c1_45__qc_lang1) ) & ~ ( v2_qc_lang1(c1_45__qc_lang1) & v4_qc_lang1(c1_45__qc_lang1) ) & ~ ( v2_qc_lang1(c1_45__qc_lang1) & v5_qc_lang1(c1_45__qc_lang1) ) & ~ ( v3_qc_lang1(c1_45__qc_lang1) & v4_qc_lang1(c1_45__qc_lang1) ) & ~ ( v3_qc_lang1(c1_45__qc_lang1) & v5_qc_lang1(c1_45__qc_lang1) ) & ~ ( v4_qc_lang1(c1_45__qc_lang1) & v5_qc_lang1(c1_45__qc_lang1) ) ), inference(conclusion,[status(thm),assumptions([dt_c1_45__qc_lang1])],[e3_45__qc_lang1,i3_45__qc_lang1]), [interesting(0.8),file(qc_lang1,i2_45__qc_lang1),[file(qc_lang1,i2_45__qc_lang1)]]). fof(i2_45_tmp__qc_lang1,plain, ( m1_subset_1(c1_45__qc_lang1,k8_qc_lang1) => ( ~ ( v2_qc_lang1(c1_45__qc_lang1) & v3_qc_lang1(c1_45__qc_lang1) ) & ~ ( v2_qc_lang1(c1_45__qc_lang1) & v4_qc_lang1(c1_45__qc_lang1) ) & ~ ( v2_qc_lang1(c1_45__qc_lang1) & v5_qc_lang1(c1_45__qc_lang1) ) & ~ ( v3_qc_lang1(c1_45__qc_lang1) & v4_qc_lang1(c1_45__qc_lang1) ) & ~ ( v3_qc_lang1(c1_45__qc_lang1) & v5_qc_lang1(c1_45__qc_lang1) ) & ~ ( v4_qc_lang1(c1_45__qc_lang1) & v5_qc_lang1(c1_45__qc_lang1) ) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_45__qc_lang1])],[dt_c1_45__qc_lang1,i2_45__qc_lang1]), [interesting(0.8),i1_45__qc_lang1]). fof(i1_45__qc_lang1,plain,( ! [A] : ( m1_subset_1(A,k8_qc_lang1) => ( ~ ( v2_qc_lang1(A) & v3_qc_lang1(A) ) & ~ ( v2_qc_lang1(A) & v4_qc_lang1(A) ) & ~ ( v2_qc_lang1(A) & v5_qc_lang1(A) ) & ~ ( v3_qc_lang1(A) & v4_qc_lang1(A) ) & ~ ( v3_qc_lang1(A) & v5_qc_lang1(A) ) & ~ ( v4_qc_lang1(A) & v5_qc_lang1(A) ) ) ) ), inference(let,[status(thm),assumptions([])],[i2_45_tmp__qc_lang1,dh_c1_45__qc_lang1]), [interesting(0.8),file(qc_lang1,i1_45__qc_lang1),[file(qc_lang1,i1_45__qc_lang1)]]). fof(t51_qc_lang1,theorem, ( ~ v2_qc_lang1(k11_qc_lang1) & ~ v3_qc_lang1(k11_qc_lang1) & ~ v4_qc_lang1(k11_qc_lang1) & ~ v5_qc_lang1(k11_qc_lang1) & ! [A] : ( m1_subset_1(A,k8_qc_lang1) => ( ~ ( v2_qc_lang1(A) & v3_qc_lang1(A) ) & ~ ( v2_qc_lang1(A) & v4_qc_lang1(A) ) & ~ ( v2_qc_lang1(A) & v5_qc_lang1(A) ) & ~ ( v3_qc_lang1(A) & v4_qc_lang1(A) ) & ~ ( v3_qc_lang1(A) & v5_qc_lang1(A) ) & ~ ( v4_qc_lang1(A) & v5_qc_lang1(A) ) ) ) ), inference(conclusion,[status(thm),assumptions([])],[e1_45__qc_lang1,i1_45__qc_lang1]), [interesting(1),file(qc_lang1,t51_qc_lang1),[file(qc_lang1,t51_qc_lang1)]]).