% Mizar problem: t12_integra5,integra5,587,35 fof(t12_integra5,conjecture,( ! [A] : ( ( v1_integra1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) ) => ! [B,C] : ( ( v1_funct_1(C) & m2_relset_1(C,k1_numbers,k1_numbers) ) => ! [D] : ( m3_integra1(D,A,k8_integra1(A)) => ( ( r1_tarski(A,B) & r2_fdiff_1(C,B) & r3_rfunct_1(k2_fdiff_1(C,B),A) ) => ( r1_xreal_0(k7_integra1(A,k1_integra5(A,k2_fdiff_1(C,B)),k8_integra1(A),D),k10_binop_2(k2_seq_1(k1_numbers,k1_numbers,C,k3_pscomp_1(A)),k2_seq_1(k1_numbers,k1_numbers,C,k4_pscomp_1(A)))) & r1_xreal_0(k10_binop_2(k2_seq_1(k1_numbers,k1_numbers,C,k3_pscomp_1(A)),k2_seq_1(k1_numbers,k1_numbers,C,k4_pscomp_1(A))),k6_integra1(A,k1_integra5(A,k2_fdiff_1(C,B)),k8_integra1(A),D)) ) ) ) ) ) ), inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_integra1,cc1_membered,cc1_relset_1,cc1_seq_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_integra1,cc2_membered,cc2_xreal_0,cc3_arytm_3,cc3_membered,cc3_xreal_0,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,cc9_membered,commutativity_k11_binop_2,commutativity_k1_nat_1,commutativity_k23_binop_2,commutativity_k2_tarski,commutativity_k2_xcmplx_0,commutativity_k3_xboole_0,commutativity_k3_xcmplx_0,commutativity_k5_subset_1,commutativity_k9_binop_2,connectedness_r1_xreal_0,d10_integra1,d10_rfunct_1,d11_rfunct_1,d1_integra5,d1_nat_2,d1_tarski,d2_integra1,d2_tarski,d3_finseq_1,d3_seq_4,d4_finseq_4,d4_seq_4,d5_funct_1,d5_integra1,d5_real_1,d5_seq_4,d6_integra1,d7_finseq_1,d7_integra1,d8_fdiff_1,d8_finseq_1,d8_integra1,d9_integra1,d9_rfunct_1,dt_k10_binop_2,dt_k11_binop_2,dt_k12_binop_2,dt_k12_finseq_1,dt_k14_integra1,dt_k15_integra1,dt_k15_rvsum_1,dt_k1_card_1,dt_k1_fdiff_1,dt_k1_finseq_1,dt_k1_funct_1,dt_k1_goboard1,dt_k1_integra1,dt_k1_integra5,dt_k1_nat_1,dt_k1_numbers,dt_k1_rcomp_1,dt_k1_relat_1,dt_k1_seq_1,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k23_binop_2,dt_k2_fdiff_1,dt_k2_finseq_1,dt_k2_integra1,dt_k2_partfun1,dt_k2_rcomp_1,dt_k2_relat_1,dt_k2_seq_1,dt_k2_seq_4,dt_k2_tarski,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k3_finseq_1,dt_k3_integra1,dt_k3_pscomp_1,dt_k3_seq_4,dt_k3_xboole_0,dt_k3_xcmplx_0,dt_k4_finseq_1,dt_k4_finseq_2,dt_k4_finseq_4,dt_k4_integra1,dt_k4_pscomp_1,dt_k4_relset_1,dt_k5_finseq_1,dt_k5_integra1,dt_k5_numbers,dt_k5_ordinal2,dt_k5_real_1,dt_k5_relset_1,dt_k5_subset_1,dt_k6_integra1,dt_k6_real_1,dt_k6_xcmplx_0,dt_k7_finseq_1,dt_k7_integra1,dt_k7_relat_1,dt_k7_xcmplx_0,dt_k8_finseq_1,dt_k8_integra1,dt_k9_binop_2,dt_k9_finseq_1,dt_m1_finseq_1,dt_m1_finseq_2,dt_m1_integra1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_finseq_2,dt_m2_integra1,dt_m2_relset_1,dt_m2_subset_1,dt_m3_integra1,existence_m1_finseq_1,existence_m1_finseq_2,existence_m1_integra1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_finseq_2,existence_m2_integra1,existence_m2_relset_1,existence_m2_subset_1,existence_m3_integra1,fc10_membered,fc10_xreal_0,fc11_membered,fc11_xreal_0,fc12_membered,fc12_xreal_0,fc13_membered,fc14_membered,fc15_membered,fc15_xreal_0,fc16_membered,fc16_xreal_0,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc1_finseq_2,fc1_integra1,fc1_ordinal2,fc1_rcomp_1,fc1_seq_1,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc23_xreal_0,fc24_xreal_0,fc27_membered,fc27_xreal_0,fc28_membered,fc28_xreal_0,fc29_membered,fc29_xreal_0,fc2_membered,fc2_rcomp_1,fc2_seq_1,fc30_membered,fc30_xreal_0,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc3_xreal_0,fc4_xreal_0,fc5_membered,fc5_xreal_0,fc6_membered,fc6_xreal_0,fc7_membered,fc7_xreal_0,fc8_membered,fc8_xreal_0,fc9_membered,fc9_xreal_0,idempotence_k3_xboole_0,idempotence_k5_subset_1,rc1_arytm_3,rc1_finseq_2,rc1_integra1,rc1_membered,rc1_rcomp_1,rc1_seq_1,rc1_xreal_0,rc2_integra1,rc2_xreal_0,rc3_integra1,rc3_xreal_0,rc4_xreal_0,redefinition_k10_binop_2,redefinition_k11_binop_2,redefinition_k12_binop_2,redefinition_k12_finseq_1,redefinition_k14_integra1,redefinition_k15_integra1,redefinition_k1_goboard1,redefinition_k1_nat_1,redefinition_k1_seq_1,redefinition_k23_binop_2,redefinition_k2_finseq_1,redefinition_k2_partfun1,redefinition_k2_seq_1,redefinition_k3_finseq_1,redefinition_k3_pscomp_1,redefinition_k4_finseq_1,redefinition_k4_pscomp_1,redefinition_k4_relset_1,redefinition_k5_numbers,redefinition_k5_real_1,redefinition_k5_relset_1,redefinition_k5_subset_1,redefinition_k6_real_1,redefinition_k8_finseq_1,redefinition_k8_integra1,redefinition_k9_binop_2,redefinition_k9_finseq_1,redefinition_m2_finseq_1,redefinition_m2_finseq_2,redefinition_m2_relset_1,redefinition_m2_subset_1,redefinition_m3_integra1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,s1_finseq_2__e11_15__integra5,s1_finseq_2__e13_15__integra5,s1_finseq_2__e7_15__integra5,spc0_boole,spc1_boole,spc1_numerals,spc2_boole,spc2_numerals,t104_rvsum_1,t106_rvsum_1,t10_finseq_2,t10_integra1,t110_finseq_2,t112_rvsum_1,t11_integra1,t11_xreal_1,t15_rcomp_1,t16_xboole_1,t17_fcont_1,t18_integra4,t19_integra4,t1_arithm,t1_integra2,t1_numerals,t1_real,t1_subset,t1_xboole_1,t21_xreal_1,t22_xreal_1,t24_integra1,t25_finseq_1,t28_nat_1,t28_partfun1,t28_xboole_1,t29_nat_1,t2_arithm,t2_boole,t2_real,t2_subset,t31_nat_2,t33_fdiff_1,t34_fdiff_1,t35_finseq_1,t36_finseq_1,t38_nat_1,t39_nat_1,t3_arithm,t3_finseq_1,t3_real,t3_rolle,t3_subset,t47_finseq_1,t4_arithm,t4_finseq_1,t4_integra1,t4_real,t4_subset,t52_xreal_1,t56_finseq_1,t57_finseq_1,t5_arithm,t5_finseq_1,t5_integra1,t5_real,t5_subset,t66_xreal_1,t68_funct_1,t6_arithm,t6_boole,t6_real,t7_boole,t7_finseq_1,t7_real,t88_xcmplx_1,t8_boole,t8_real,t8_xreal_1,t9_xreal_1]), [file(integra5,t12_integra5)]). fof(antisymmetry_r2_hidden,axiom,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), []). fof(cc10_membered,axiom,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,A) => v1_xcmplx_0(B) ) ) ), file(membered,cc10_membered), []). fof(cc11_membered,axiom,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) ) ) ) ), file(membered,cc11_membered), []). fof(cc12_membered,axiom,( ! [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), []). fof(cc13_membered,axiom,( ! [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), []). fof(cc14_membered,axiom,( ! [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), []). fof(cc15_membered,axiom,( ! [A] : ( v1_xboole_0(A) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc15_membered), []). fof(cc16_membered,axiom,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_membered(B) ) ) ), file(membered,cc16_membered), []). fof(cc17_membered,axiom,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) ) ) ) ), file(membered,cc17_membered), []). fof(cc18_membered,axiom,( ! [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), []). fof(cc19_membered,axiom,( ! [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), []). fof(cc1_arytm_3,axiom,( ! [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), []). fof(cc1_integra1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( v1_integra1(A) => ( ~ v1_xboole_0(A) & v1_membered(A) & v2_membered(A) & v1_rcomp_1(A) ) ) ) ), file(integra1,cc1_integra1), []). fof(cc1_membered,axiom,( ! [A] : ( v5_membered(A) => v4_membered(A) ) ), file(membered,cc1_membered), []). fof(cc1_relset_1,axiom,( ! [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), []). fof(cc1_seq_1,axiom,( ! [A,B] : ( v2_membered(B) => ! [C] : ( m1_relset_1(C,A,B) => ( v1_funct_1(C) => ( v1_funct_1(C) & v1_seq_1(C) ) ) ) ) ), file(seq_1,cc1_seq_1), []). fof(cc1_xreal_0,axiom,( ! [A] : ( v4_ordinal2(A) => v1_xreal_0(A) ) ), file(xreal_0,cc1_xreal_0), []). fof(cc20_membered,axiom,( ! [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), []). fof(cc2_arytm_3,axiom,( ! [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), []). fof(cc2_integra1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( v1_integra1(A) => ( v1_membered(A) & v2_membered(A) & v3_seq_4(A) ) ) ) ), file(integra1,cc2_integra1), []). fof(cc2_membered,axiom,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), []). fof(cc2_xreal_0,axiom,( ! [A] : ( v1_xreal_0(A) => v1_xcmplx_0(A) ) ), file(xreal_0,cc2_xreal_0), []). fof(cc3_arytm_3,axiom,( ! [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), []). fof(cc3_membered,axiom,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), []). fof(cc3_xreal_0,axiom,( ! [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), []). fof(cc4_membered,axiom,( ! [A] : ( v2_membered(A) => v1_membered(A) ) ), file(membered,cc4_membered), []). fof(cc4_xreal_0,axiom,( ! [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), []). fof(cc5_xreal_0,axiom,( ! [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), []). fof(cc6_membered,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( v1_membered(A) & v2_membered(A) ) ) ), file(membered,cc6_membered), []). fof(cc6_xreal_0,axiom,( ! [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), []). fof(cc7_xreal_0,axiom,( ! [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), []). fof(cc8_xreal_0,axiom,( ! [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), []). fof(cc9_membered,axiom,( ! [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), []). fof(commutativity_k11_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k11_binop_2(A,B) = k11_binop_2(B,A) ) ), file(binop_2,k11_binop_2), []). fof(commutativity_k1_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k1_nat_1(A,B) = k1_nat_1(B,A) ) ), file(nat_1,k1_nat_1), []). fof(commutativity_k23_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k23_binop_2(A,B) = k23_binop_2(B,A) ) ), file(binop_2,k23_binop_2), []). fof(commutativity_k2_tarski,axiom,( ! [A,B] : k2_tarski(A,B) = k2_tarski(B,A) ), file(tarski,k2_tarski), []). fof(commutativity_k2_xcmplx_0,axiom,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k2_xcmplx_0(A,B) = k2_xcmplx_0(B,A) ) ), file(xcmplx_0,k2_xcmplx_0), []). fof(commutativity_k3_xboole_0,axiom,( ! [A,B] : k3_xboole_0(A,B) = k3_xboole_0(B,A) ), file(xboole_0,k3_xboole_0), []). fof(commutativity_k3_xcmplx_0,axiom,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k3_xcmplx_0(A,B) = k3_xcmplx_0(B,A) ) ), file(xcmplx_0,k3_xcmplx_0), []). fof(commutativity_k5_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k5_subset_1(A,B,C) = k5_subset_1(A,C,B) ) ), file(subset_1,k5_subset_1), []). fof(commutativity_k9_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k9_binop_2(A,B) = k9_binop_2(B,A) ) ), file(binop_2,k9_binop_2), []). fof(connectedness_r1_xreal_0,axiom,( ! [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), []). fof(d10_integra1,axiom,( ! [A] : ( ( v1_integra1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) ) => ! [B] : ( ( v1_funct_1(B) & m2_relset_1(B,A,k1_numbers) ) => ! [C] : ( ( ~ v1_xboole_0(C) & m2_integra1(C,A) ) => ! [D] : ( m3_integra1(D,A,C) => k7_integra1(A,B,C,D) = k15_rvsum_1(k5_integra1(A,B,C,D)) ) ) ) ) ), file(integra1,d10_integra1), []). fof(d10_rfunct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) ) => ! [B] : ( r2_rfunct_1(A,B) <=> ? [C] : ( v1_xreal_0(C) & ! [D] : ( r2_hidden(D,k3_xboole_0(B,k1_relat_1(A))) => r1_xreal_0(C,k1_seq_1(A,D)) ) ) ) ) ), file(rfunct_1,d10_rfunct_1), []). fof(d11_rfunct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) ) => ! [B] : ( r3_rfunct_1(A,B) <=> ( r1_rfunct_1(A,B) & r2_rfunct_1(A,B) ) ) ) ), file(rfunct_1,d11_rfunct_1), []). fof(d1_integra5,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) ) => ! [B] : ( ( v1_funct_1(B) & m2_relset_1(B,k1_numbers,k1_numbers) ) => k1_integra5(A,B) = k2_partfun1(k1_numbers,k1_numbers,B,A) ) ) ), file(integra5,d1_integra5), []). fof(d1_nat_2,axiom,( ! [A] : ( v4_ordinal2(A) => ( v1_realset1(A) <=> ( A = 0 | A = 1 ) ) ) ), file(nat_2,d1_nat_2), []). fof(d1_tarski,axiom,( ! [A,B] : ( B = k1_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> C = A ) ) ), file(tarski,d1_tarski), []). fof(d2_integra1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_rcomp_1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) ) => ! [B] : ( ( ~ v1_xboole_0(B) & v1_goboard1(B) & m2_finseq_1(B,k1_numbers) ) => ( m1_integra1(B,A) <=> ( r1_tarski(k5_relset_1(k5_numbers,k1_numbers,B),A) & k1_goboard1(B,k3_finseq_1(B)) = k3_pscomp_1(A) ) ) ) ) ), file(integra1,d2_integra1), []). fof(d2_tarski,axiom,( ! [A,B,C] : ( C = k2_tarski(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( D = A | D = B ) ) ) ), file(tarski,d2_tarski), []). fof(d3_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( B = k3_finseq_1(A) <=> k2_finseq_1(B) = k1_relat_1(A) ) ) ) ), file(finseq_1,d3_finseq_1), []). fof(d3_seq_4,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( v3_seq_4(A) <=> ( v2_seq_4(A) & v1_seq_4(A) ) ) ) ), file(seq_4,d3_seq_4), []). fof(d4_finseq_4,axiom,( ! [A,B,C] : ( ( v1_funct_1(C) & m2_relset_1(C,A,B) ) => ! [D] : ( r2_hidden(D,k1_relat_1(C)) => k4_finseq_4(A,B,C,D) = k1_funct_1(C,D) ) ) ), file(finseq_4,d4_finseq_4), []). fof(d4_seq_4,axiom,( ! [A] : ( v2_membered(A) => ( v1_seq_4(A) => ( v1_xboole_0(A) | ! [B] : ( v1_xreal_0(B) => ( B = k2_seq_4(A) <=> ( ! [C] : ( v1_xreal_0(C) => ( r2_hidden(C,A) => r1_xreal_0(C,B) ) ) & ! [C] : ( v1_xreal_0(C) => ~ ( ~ r1_xreal_0(C,0) & ! [D] : ( v1_xreal_0(D) => ~ ( r2_hidden(D,A) & ~ r1_xreal_0(D,k6_xcmplx_0(B,C)) ) ) ) ) ) ) ) ) ) ) ), file(seq_4,d4_seq_4), []). fof(d5_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( B = k2_relat_1(A) <=> ! [C] : ( r2_hidden(C,B) <=> ? [D] : ( r2_hidden(D,k1_relat_1(A)) & C = k1_funct_1(A,D) ) ) ) ) ), file(funct_1,d5_funct_1), []). fof(d5_integra1,axiom,( ! [A] : ( ( v1_integra1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) ) => ! [B] : ( ( ~ v1_xboole_0(B) & m2_integra1(B,A) ) => ! [C] : ( m3_integra1(C,A,B) => ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( r2_hidden(D,k4_finseq_1(C)) => ! [E] : ( ( v1_integra1(E) & m1_subset_1(E,k1_zfmisc_1(k1_numbers)) ) => ( ( D = 1 => ( E = k2_integra1(A,B,C,D) <=> ( k4_pscomp_1(E) = k4_pscomp_1(A) & k3_pscomp_1(E) = k1_goboard1(C,D) ) ) ) & ( D != 1 => ( E = k2_integra1(A,B,C,D) <=> ( k4_pscomp_1(E) = k2_seq_1(k5_numbers,k1_numbers,C,k10_binop_2(D,1)) & k3_pscomp_1(E) = k1_goboard1(C,D) ) ) ) ) ) ) ) ) ) ) ), file(integra1,d5_integra1), []). fof(d5_real_1,axiom,( ! [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), []). fof(d5_seq_4,axiom,( ! [A] : ( v2_membered(A) => ( v2_seq_4(A) => ( v1_xboole_0(A) | ! [B] : ( v1_xreal_0(B) => ( B = k3_seq_4(A) <=> ( ! [C] : ( v1_xreal_0(C) => ( r2_hidden(C,A) => r1_xreal_0(B,C) ) ) & ! [C] : ( v1_xreal_0(C) => ~ ( ~ r1_xreal_0(C,0) & ! [D] : ( v1_xreal_0(D) => ~ ( r2_hidden(D,A) & ~ r1_xreal_0(k2_xcmplx_0(B,C),D) ) ) ) ) ) ) ) ) ) ) ), file(seq_4,d5_seq_4), []). fof(d6_integra1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => k3_integra1(A) = k10_binop_2(k3_pscomp_1(A),k4_pscomp_1(A)) ) ), file(integra1,d6_integra1), []). fof(d7_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v1_finseq_1(C) ) => ( C = k7_finseq_1(A,B) <=> ( k4_finseq_1(C) = k2_finseq_1(k1_nat_1(k3_finseq_1(A),k3_finseq_1(B))) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( r2_hidden(D,k4_finseq_1(A)) => k1_funct_1(C,D) = k1_funct_1(A,D) ) ) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( r2_hidden(D,k4_finseq_1(B)) => k1_funct_1(C,k1_nat_1(k3_finseq_1(A),D)) = k1_funct_1(B,D) ) ) ) ) ) ) ) ), file(finseq_1,d7_finseq_1), []). fof(d7_integra1,axiom,( ! [A] : ( ( v1_integra1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) ) => ! [B] : ( ( v1_funct_1(B) & m2_relset_1(B,A,k1_numbers) ) => ! [C] : ( ( ~ v1_xboole_0(C) & m2_integra1(C,A) ) => ! [D] : ( m3_integra1(D,A,C) => ! [E] : ( m2_finseq_1(E,k1_numbers) => ( E = k4_integra1(A,B,C,D) <=> ( k3_finseq_1(E) = k3_finseq_1(D) & ! [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) => ( r2_hidden(F,k2_finseq_1(k3_finseq_1(D))) => k1_goboard1(E,F) = k11_binop_2(k3_pscomp_1(k5_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,k2_integra1(A,C,D,F)))),k3_integra1(k2_integra1(A,C,D,F))) ) ) ) ) ) ) ) ) ) ), file(integra1,d7_integra1), []). fof(d8_fdiff_1,axiom,( ! [A] : ( ( v1_funct_1(A) & m2_relset_1(A,k1_numbers,k1_numbers) ) => ! [B] : ( r2_fdiff_1(A,B) => ! [C] : ( ( v1_funct_1(C) & m2_relset_1(C,k1_numbers,k1_numbers) ) => ( C = k2_fdiff_1(A,B) <=> ( k4_relset_1(k1_numbers,k1_numbers,C) = B & ! [D] : ( m1_subset_1(D,k1_numbers) => ( r2_hidden(D,B) => k2_seq_1(k1_numbers,k1_numbers,C,D) = k1_fdiff_1(A,D) ) ) ) ) ) ) ) ), file(fdiff_1,d8_fdiff_1), []). fof(d8_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( B = k9_finseq_1(A) <=> ( k1_relat_1(B) = k2_finseq_1(1) & k1_funct_1(B,1) = A ) ) ) ), file(finseq_1,d8_finseq_1), []). fof(d8_integra1,axiom,( ! [A] : ( ( v1_integra1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) ) => ! [B] : ( ( v1_funct_1(B) & m2_relset_1(B,A,k1_numbers) ) => ! [C] : ( ( ~ v1_xboole_0(C) & m2_integra1(C,A) ) => ! [D] : ( m3_integra1(D,A,C) => ! [E] : ( m2_finseq_1(E,k1_numbers) => ( E = k5_integra1(A,B,C,D) <=> ( k3_finseq_1(E) = k3_finseq_1(D) & ! [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) => ( r2_hidden(F,k2_finseq_1(k3_finseq_1(D))) => k1_goboard1(E,F) = k11_binop_2(k4_pscomp_1(k5_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,k2_integra1(A,C,D,F)))),k3_integra1(k2_integra1(A,C,D,F))) ) ) ) ) ) ) ) ) ) ), file(integra1,d8_integra1), []). fof(d9_integra1,axiom,( ! [A] : ( ( v1_integra1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) ) => ! [B] : ( ( v1_funct_1(B) & m2_relset_1(B,A,k1_numbers) ) => ! [C] : ( ( ~ v1_xboole_0(C) & m2_integra1(C,A) ) => ! [D] : ( m3_integra1(D,A,C) => k6_integra1(A,B,C,D) = k15_rvsum_1(k4_integra1(A,B,C,D)) ) ) ) ) ), file(integra1,d9_integra1), []). fof(d9_rfunct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) ) => ! [B] : ( r1_rfunct_1(A,B) <=> ? [C] : ( v1_xreal_0(C) & ! [D] : ( r2_hidden(D,k3_xboole_0(B,k1_relat_1(A))) => r1_xreal_0(k1_seq_1(A,D),C) ) ) ) ) ), file(rfunct_1,d9_rfunct_1), []). fof(dt_k10_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k10_binop_2(A,B),k1_numbers) ) ), file(binop_2,k10_binop_2), []). fof(dt_k11_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k11_binop_2(A,B),k1_numbers) ) ), file(binop_2,k11_binop_2), []). fof(dt_k12_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k12_binop_2(A,B),k1_numbers) ) ), file(binop_2,k12_binop_2), []). fof(dt_k12_finseq_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => m2_finseq_1(k12_finseq_1(A,B),A) ) ), file(finseq_1,k12_finseq_1), []). fof(dt_k14_integra1,axiom,( ! [A,B,C,D] : ( ( v1_integra1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & v1_funct_1(B) & m1_relset_1(B,A,k1_numbers) & ~ v1_xboole_0(C) & m2_integra1(C,A) & m1_subset_1(D,C) ) => ( ~ v1_xboole_0(k14_integra1(A,B,C,D)) & m2_finseq_1(k14_integra1(A,B,C,D),k1_numbers) ) ) ), file(integra1,k14_integra1), []). fof(dt_k15_integra1,axiom,( ! [A,B,C,D] : ( ( v1_integra1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & v1_funct_1(B) & m1_relset_1(B,A,k1_numbers) & ~ v1_xboole_0(C) & m2_integra1(C,A) & m1_subset_1(D,C) ) => ( ~ v1_xboole_0(k15_integra1(A,B,C,D)) & m2_finseq_1(k15_integra1(A,B,C,D),k1_numbers) ) ) ), file(integra1,k15_integra1), []). fof(dt_k15_rvsum_1,axiom,( ! [A] : ( m1_finseq_1(A,k1_numbers) => m1_subset_1(k15_rvsum_1(A),k1_numbers) ) ), file(rvsum_1,k15_rvsum_1), []). fof(dt_k1_card_1,axiom,( ! [A] : v1_card_1(k1_card_1(A)) ), file(card_1,k1_card_1), []). fof(dt_k1_fdiff_1,axiom,( ! [A,B] : ( ( v1_funct_1(A) & m1_relset_1(A,k1_numbers,k1_numbers) & v1_xreal_0(B) ) => m1_subset_1(k1_fdiff_1(A,B),k1_numbers) ) ), file(fdiff_1,k1_fdiff_1), []). fof(dt_k1_finseq_1,axiom,( $true ), file(finseq_1,k1_finseq_1), []). fof(dt_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), []). fof(dt_k1_goboard1,axiom,( ! [A,B] : ( ( m1_finseq_1(A,k1_numbers) & m1_subset_1(B,k5_numbers) ) => m1_subset_1(k1_goboard1(A,B),k1_numbers) ) ), file(goboard1,k1_goboard1), []). fof(dt_k1_integra1,axiom,( $true ), file(integra1,k1_integra1), []). fof(dt_k1_integra5,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & v1_funct_1(B) & m1_relset_1(B,k1_numbers,k1_numbers) ) => ( v1_funct_1(k1_integra5(A,B)) & m2_relset_1(k1_integra5(A,B),A,k1_numbers) ) ) ), file(integra5,k1_integra5), []). fof(dt_k1_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k1_nat_1(A,B),k1_numbers,k5_numbers) ) ), file(nat_1,k1_nat_1), []). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), []). fof(dt_k1_rcomp_1,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => m1_subset_1(k1_rcomp_1(A,B),k1_zfmisc_1(k1_numbers)) ) ), file(rcomp_1,k1_rcomp_1), []). fof(dt_k1_relat_1,axiom,( $true ), file(relat_1,k1_relat_1), []). fof(dt_k1_seq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) ) => m1_subset_1(k1_seq_1(A,B),k1_numbers) ) ), file(seq_1,k1_seq_1), []). fof(dt_k1_tarski,axiom,( $true ), file(tarski,k1_tarski), []). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), []). fof(dt_k1_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), []). fof(dt_k23_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k23_binop_2(A,B),k1_numbers,k5_numbers) ) ), file(binop_2,k23_binop_2), []). fof(dt_k2_fdiff_1,axiom,( ! [A,B] : ( ( v1_funct_1(A) & m1_relset_1(A,k1_numbers,k1_numbers) ) => ( v1_funct_1(k2_fdiff_1(A,B)) & m2_relset_1(k2_fdiff_1(A,B),k1_numbers,k1_numbers) ) ) ), file(fdiff_1,k2_fdiff_1), []). 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), []). fof(dt_k2_integra1,axiom,( ! [A,B,C,D] : ( ( v1_integra1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(B) & m2_integra1(B,A) & m1_subset_1(C,B) & m1_subset_1(D,k5_numbers) ) => ( v1_integra1(k2_integra1(A,B,C,D)) & m1_subset_1(k2_integra1(A,B,C,D),k1_zfmisc_1(k1_numbers)) ) ) ), file(integra1,k2_integra1), []). fof(dt_k2_partfun1,axiom,( ! [A,B,C,D] : ( ( v1_funct_1(C) & m1_relset_1(C,A,B) ) => ( v1_funct_1(k2_partfun1(A,B,C,D)) & m2_relset_1(k2_partfun1(A,B,C,D),A,B) ) ) ), file(partfun1,k2_partfun1), []). fof(dt_k2_rcomp_1,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => m1_subset_1(k2_rcomp_1(A,B),k1_zfmisc_1(k1_numbers)) ) ), file(rcomp_1,k2_rcomp_1), []). fof(dt_k2_relat_1,axiom,( $true ), file(relat_1,k2_relat_1), []). fof(dt_k2_seq_1,axiom,( ! [A,B,C,D] : ( ( v2_membered(B) & v1_funct_1(C) & m1_relset_1(C,A,B) ) => m1_subset_1(k2_seq_1(A,B,C,D),k1_numbers) ) ), file(seq_1,k2_seq_1), []). fof(dt_k2_seq_4,axiom,( ! [A] : ( v2_membered(A) => v1_xreal_0(k2_seq_4(A)) ) ), file(seq_4,k2_seq_4), []). fof(dt_k2_tarski,axiom,( $true ), file(tarski,k2_tarski), []). fof(dt_k2_xcmplx_0,axiom,( $true ), file(xcmplx_0,k2_xcmplx_0), []). fof(dt_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), []). fof(dt_k3_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => m2_subset_1(k3_finseq_1(A),k1_numbers,k5_numbers) ) ), file(finseq_1,k3_finseq_1), []). fof(dt_k3_integra1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => m1_subset_1(k3_integra1(A),k1_numbers) ) ), file(integra1,k3_integra1), []). fof(dt_k3_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => m1_subset_1(k3_pscomp_1(A),k1_numbers) ) ), file(pscomp_1,k3_pscomp_1), []). fof(dt_k3_seq_4,axiom,( ! [A] : ( v2_membered(A) => v1_xreal_0(k3_seq_4(A)) ) ), file(seq_4,k3_seq_4), []). fof(dt_k3_xboole_0,axiom,( $true ), file(xboole_0,k3_xboole_0), []). fof(dt_k3_xcmplx_0,axiom,( $true ), file(xcmplx_0,k3_xcmplx_0), []). fof(dt_k4_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => m1_subset_1(k4_finseq_1(A),k1_zfmisc_1(k5_numbers)) ) ), file(finseq_1,k4_finseq_1), []). fof(dt_k4_finseq_2,axiom,( ! [A,B] : ( v4_ordinal2(A) => m1_finseq_2(k4_finseq_2(A,B),B) ) ), file(finseq_2,k4_finseq_2), []). fof(dt_k4_finseq_4,axiom,( ! [A,B,C,D] : ( ( v1_funct_1(C) & m1_relset_1(C,A,B) ) => m1_subset_1(k4_finseq_4(A,B,C,D),B) ) ), file(finseq_4,k4_finseq_4), []). fof(dt_k4_integra1,axiom,( ! [A,B,C,D] : ( ( v1_integra1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & v1_funct_1(B) & m1_relset_1(B,A,k1_numbers) & ~ v1_xboole_0(C) & m2_integra1(C,A) & m1_subset_1(D,C) ) => m2_finseq_1(k4_integra1(A,B,C,D),k1_numbers) ) ), file(integra1,k4_integra1), []). fof(dt_k4_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => m1_subset_1(k4_pscomp_1(A),k1_numbers) ) ), file(pscomp_1,k4_pscomp_1), []). fof(dt_k4_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => m1_subset_1(k4_relset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(relset_1,k4_relset_1), []). fof(dt_k5_finseq_1,axiom,( $true ), file(finseq_1,k5_finseq_1), []). fof(dt_k5_integra1,axiom,( ! [A,B,C,D] : ( ( v1_integra1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & v1_funct_1(B) & m1_relset_1(B,A,k1_numbers) & ~ v1_xboole_0(C) & m2_integra1(C,A) & m1_subset_1(D,C) ) => m2_finseq_1(k5_integra1(A,B,C,D),k1_numbers) ) ), file(integra1,k5_integra1), []). fof(dt_k5_numbers,axiom,( m1_subset_1(k5_numbers,k1_zfmisc_1(k1_numbers)) ), file(numbers,k5_numbers), []). fof(dt_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), []). fof(dt_k5_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k5_real_1(A,B),k1_numbers) ) ), file(real_1,k5_real_1), []). fof(dt_k5_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => m1_subset_1(k5_relset_1(A,B,C),k1_zfmisc_1(B)) ) ), file(relset_1,k5_relset_1), []). fof(dt_k5_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => m1_subset_1(k5_subset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(subset_1,k5_subset_1), []). fof(dt_k6_integra1,axiom,( ! [A,B,C,D] : ( ( v1_integra1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & v1_funct_1(B) & m1_relset_1(B,A,k1_numbers) & ~ v1_xboole_0(C) & m2_integra1(C,A) & m1_subset_1(D,C) ) => m1_subset_1(k6_integra1(A,B,C,D),k1_numbers) ) ), file(integra1,k6_integra1), []). fof(dt_k6_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k6_real_1(A,B),k1_numbers) ) ), file(real_1,k6_real_1), []). fof(dt_k6_xcmplx_0,axiom,( $true ), file(xcmplx_0,k6_xcmplx_0), []). fof(dt_k7_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( v1_relat_1(k7_finseq_1(A,B)) & v1_funct_1(k7_finseq_1(A,B)) & v1_finseq_1(k7_finseq_1(A,B)) ) ) ), file(finseq_1,k7_finseq_1), []). fof(dt_k7_integra1,axiom,( ! [A,B,C,D] : ( ( v1_integra1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & v1_funct_1(B) & m1_relset_1(B,A,k1_numbers) & ~ v1_xboole_0(C) & m2_integra1(C,A) & m1_subset_1(D,C) ) => m1_subset_1(k7_integra1(A,B,C,D),k1_numbers) ) ), file(integra1,k7_integra1), []). fof(dt_k7_relat_1,axiom,( ! [A,B] : ( v1_relat_1(A) => v1_relat_1(k7_relat_1(A,B)) ) ), file(relat_1,k7_relat_1), []). fof(dt_k7_xcmplx_0,axiom,( $true ), file(xcmplx_0,k7_xcmplx_0), []). fof(dt_k8_finseq_1,axiom,( ! [A,B,C] : ( ( m1_finseq_1(B,A) & m1_finseq_1(C,A) ) => m2_finseq_1(k8_finseq_1(A,B,C),A) ) ), file(finseq_1,k8_finseq_1), []). fof(dt_k8_integra1,axiom,( ! [A] : ( ( v1_integra1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) ) => m2_integra1(k8_integra1(A),A) ) ), file(integra1,k8_integra1), []). fof(dt_k9_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k9_binop_2(A,B),k1_numbers) ) ), file(binop_2,k9_binop_2), []). 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), []). 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), []). fof(dt_m1_finseq_2,axiom,( $true ), file(finseq_2,m1_finseq_2), []). fof(dt_m1_integra1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_rcomp_1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) ) => ! [B] : ( m1_integra1(B,A) => ( ~ v1_xboole_0(B) & v1_goboard1(B) & m2_finseq_1(B,k1_numbers) ) ) ) ), file(integra1,m1_integra1), []). fof(dt_m1_relset_1,axiom,( $true ), file(relset_1,m1_relset_1), []). fof(dt_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_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), []). fof(dt_m2_finseq_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & m1_finseq_2(B,A) ) => ! [C] : ( m2_finseq_2(C,A,B) => m2_finseq_1(C,A) ) ) ), file(finseq_2,m2_finseq_2), []). fof(dt_m2_integra1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_rcomp_1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) ) => ! [B] : ( m2_integra1(B,A) => ~ v1_xboole_0(B) ) ) ), file(integra1,m2_integra1), []). 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), []). 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), []). fof(dt_m3_integra1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_rcomp_1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(B) & m2_integra1(B,A) ) => ! [C] : ( m3_integra1(C,A,B) => m1_integra1(C,A) ) ) ), file(integra1,m3_integra1), []). fof(existence_m1_finseq_1,axiom,( ! [A] : ? [B] : m1_finseq_1(B,A) ), file(finseq_1,m1_finseq_1), []). fof(existence_m1_finseq_2,axiom,( ! [A] : ? [B] : m1_finseq_2(B,A) ), file(finseq_2,m1_finseq_2), []). fof(existence_m1_integra1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_rcomp_1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) ) => ? [B] : m1_integra1(B,A) ) ), file(integra1,m1_integra1), []). fof(existence_m1_relset_1,axiom,( ! [A,B] : ? [C] : m1_relset_1(C,A,B) ), file(relset_1,m1_relset_1), []). fof(existence_m1_subset_1,axiom,( ! [A] : ? [B] : m1_subset_1(B,A) ), file(subset_1,m1_subset_1), []). fof(existence_m2_finseq_1,axiom,( ! [A] : ? [B] : m2_finseq_1(B,A) ), file(finseq_1,m2_finseq_1), []). fof(existence_m2_finseq_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & m1_finseq_2(B,A) ) => ? [C] : m2_finseq_2(C,A,B) ) ), file(finseq_2,m2_finseq_2), []). fof(existence_m2_integra1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_rcomp_1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) ) => ? [B] : m2_integra1(B,A) ) ), file(integra1,m2_integra1), []). fof(existence_m2_relset_1,axiom,( ! [A,B] : ? [C] : m2_relset_1(C,A,B) ), file(relset_1,m2_relset_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), []). fof(existence_m3_integra1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_rcomp_1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(B) & m2_integra1(B,A) ) => ? [C] : m3_integra1(C,A,B) ) ), file(integra1,m3_integra1), []). fof(fc10_membered,axiom,( ! [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), []). fof(fc10_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v2_xreal_0(k2_xcmplx_0(B,A)) & ~ v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc10_xreal_0), []). fof(fc11_membered,axiom,( ! [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), []). fof(fc11_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v2_xreal_0(k2_xcmplx_0(A,B)) & v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc11_xreal_0), []). fof(fc12_membered,axiom,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => v1_membered(k2_tarski(A,B)) ) ), file(membered,fc12_membered), []). fof(fc12_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & ~ v2_xreal_0(k2_xcmplx_0(B,A)) & v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc12_xreal_0), []). fof(fc13_membered,axiom,( ! [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), []). fof(fc14_membered,axiom,( ! [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), []). fof(fc15_membered,axiom,( ! [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), []). fof(fc15_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & ~ v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc15_xreal_0), []). fof(fc16_membered,axiom,( ! [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), []). fof(fc16_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & ~ v2_xreal_0(k6_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc16_xreal_0), []). fof(fc17_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(A,B)) & v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v2_xreal_0(k6_xcmplx_0(A,B)) & ~ v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc17_xreal_0), []). fof(fc18_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(B,A)) & v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & ~ v2_xreal_0(k6_xcmplx_0(B,A)) & v3_xreal_0(k6_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc18_xreal_0), []). fof(fc19_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(A,B)) & v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & ~ v2_xreal_0(k6_xcmplx_0(A,B)) & v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc19_xreal_0), []). fof(fc1_finseq_2,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & ~ v1_xboole_0(B) ) => ~ v1_xboole_0(k4_finseq_2(A,B)) ) ), file(finseq_2,fc1_finseq_2), []). fof(fc1_integra1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_rcomp_1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) ) => ~ v1_xboole_0(k1_integra1(A)) ) ), file(integra1,fc1_integra1), []). fof(fc1_ordinal2,axiom, ( v1_ordinal1(k5_ordinal2) & v2_ordinal1(k5_ordinal2) & v3_ordinal1(k5_ordinal2) & ~ v1_xboole_0(k5_ordinal2) ), file(ordinal2,fc1_ordinal2), []). fof(fc1_rcomp_1,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_membered(k2_rcomp_1(A,B)) & v2_membered(k2_rcomp_1(A,B)) & v3_rcomp_1(k2_rcomp_1(A,B)) ) ) ), file(rcomp_1,fc1_rcomp_1), []). fof(fc1_seq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) ) => ( v1_xcmplx_0(k1_funct_1(A,B)) & v1_xreal_0(k1_funct_1(A,B)) ) ) ), file(seq_1,fc1_seq_1), []). fof(fc20_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(B,A)) & v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & v2_xreal_0(k6_xcmplx_0(B,A)) & ~ v3_xreal_0(k6_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc20_xreal_0), []). fof(fc21_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & ~ v2_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc21_xreal_0), []). fof(fc22_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(B,A)) & v1_xreal_0(k3_xcmplx_0(B,A)) & ~ v2_xreal_0(k3_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc22_xreal_0), []). fof(fc23_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & ~ v3_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc23_xreal_0), []). fof(fc24_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & ~ v3_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc24_xreal_0), []). fof(fc27_membered,axiom,( ! [A,B] : ( v1_membered(A) => v1_membered(k3_xboole_0(A,B)) ) ), file(membered,fc27_membered), []). fof(fc27_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(A,B)) & v1_xreal_0(k7_xcmplx_0(A,B)) & ~ v2_xreal_0(k7_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc27_xreal_0), []). fof(fc28_membered,axiom,( ! [A,B] : ( v1_membered(A) => v1_membered(k3_xboole_0(B,A)) ) ), file(membered,fc28_membered), []). fof(fc28_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(B,A)) & v1_xreal_0(k7_xcmplx_0(B,A)) & ~ v2_xreal_0(k7_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc28_xreal_0), []). fof(fc29_membered,axiom,( ! [A,B] : ( v2_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc29_membered), []). fof(fc29_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(A,B)) & v1_xreal_0(k7_xcmplx_0(A,B)) & ~ v3_xreal_0(k7_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc29_xreal_0), []). fof(fc2_membered,axiom, ( ~ v1_xboole_0(k1_numbers) & v1_membered(k1_numbers) & v2_membered(k1_numbers) ), file(membered,fc2_membered), []). fof(fc2_rcomp_1,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_membered(k1_rcomp_1(A,B)) & v2_membered(k1_rcomp_1(A,B)) & v2_rcomp_1(k1_rcomp_1(A,B)) ) ) ), file(rcomp_1,fc2_rcomp_1), []). fof(fc2_seq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_seq_1(A) ) => ( v1_relat_1(k7_relat_1(A,B)) & v1_seq_1(k7_relat_1(A,B)) ) ) ), file(seq_1,fc2_seq_1), []). fof(fc30_membered,axiom,( ! [A,B] : ( v2_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc30_membered), []). fof(fc30_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(A,B)) & v1_xreal_0(k7_xcmplx_0(A,B)) & ~ v3_xreal_0(k7_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc30_xreal_0), []). fof(fc31_membered,axiom,( ! [A,B] : ( v3_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) & v3_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc31_membered), []). fof(fc32_membered,axiom,( ! [A,B] : ( v3_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) & v3_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc32_membered), []). fof(fc33_membered,axiom,( ! [A,B] : ( v4_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) & v3_membered(k3_xboole_0(A,B)) & v4_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc33_membered), []). fof(fc34_membered,axiom,( ! [A,B] : ( v4_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) & v3_membered(k3_xboole_0(B,A)) & v4_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc34_membered), []). fof(fc35_membered,axiom,( ! [A,B] : ( v5_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) & v3_membered(k3_xboole_0(A,B)) & v4_membered(k3_xboole_0(A,B)) & v5_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc35_membered), []). fof(fc36_membered,axiom,( ! [A,B] : ( v5_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) & v3_membered(k3_xboole_0(B,A)) & v4_membered(k3_xboole_0(B,A)) & v5_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc36_membered), []). fof(fc3_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc3_xreal_0), []). fof(fc4_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc4_xreal_0), []). fof(fc5_membered,axiom, ( v1_membered(k5_ordinal2) & v2_membered(k5_ordinal2) & v3_membered(k5_ordinal2) & v4_membered(k5_ordinal2) & v5_membered(k5_ordinal2) ), file(membered,fc5_membered), []). fof(fc5_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc5_xreal_0), []). fof(fc6_membered,axiom, ( 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), []). fof(fc6_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(A,B)) & v1_xreal_0(k7_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc6_xreal_0), []). fof(fc7_membered,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_membered(k1_tarski(A)) ) ), file(membered,fc7_membered), []). fof(fc7_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc7_xreal_0), []). fof(fc8_membered,axiom,( ! [A] : ( v1_xreal_0(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) ) ) ), file(membered,fc8_membered), []). fof(fc8_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v2_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc8_xreal_0), []). fof(fc9_membered,axiom,( ! [A] : ( v1_rat_1(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) & v3_membered(k1_tarski(A)) ) ) ), file(membered,fc9_membered), []). fof(fc9_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v2_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc9_xreal_0), []). fof(idempotence_k3_xboole_0,axiom,( ! [A,B] : k3_xboole_0(A,A) = A ), file(xboole_0,k3_xboole_0), []). fof(idempotence_k5_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k5_subset_1(A,B,B) = B ) ), file(subset_1,k5_subset_1), []). fof(rc1_arytm_3,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ), file(arytm_3,rc1_arytm_3), []). fof(rc1_finseq_2,axiom,( ! [A] : ? [B] : ( m1_finseq_2(B,A) & ~ v1_xboole_0(B) ) ), file(finseq_2,rc1_finseq_2), []). fof(rc1_integra1,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & v1_membered(A) & v2_membered(A) & v1_integra1(A) ) ), file(integra1,rc1_integra1), []). fof(rc1_membered,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ), file(membered,rc1_membered), []). fof(rc1_rcomp_1,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & v1_membered(A) & v2_membered(A) & v3_rcomp_1(A) ) ), file(rcomp_1,rc1_rcomp_1), []). fof(rc1_seq_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) ) ), file(seq_1,rc1_seq_1), []). fof(rc1_xreal_0,axiom,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(xreal_0,rc1_xreal_0), []). fof(rc2_integra1,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(A) & v1_membered(A) & v2_membered(A) & v3_seq_4(A) & v1_rcomp_1(A) & v1_integra1(A) ) ), file(integra1,rc2_integra1), []). fof(rc2_xreal_0,axiom,( ? [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), []). fof(rc3_integra1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_rcomp_1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) ) => ? [B] : ( m2_integra1(B,A) & ~ v1_xboole_0(B) ) ) ), file(integra1,rc3_integra1), []). fof(rc3_xreal_0,axiom,( ? [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), []). fof(rc4_xreal_0,axiom,( ? [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), []). fof(redefinition_k10_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k10_binop_2(A,B) = k6_xcmplx_0(A,B) ) ), file(binop_2,k10_binop_2), []). fof(redefinition_k11_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k11_binop_2(A,B) = k3_xcmplx_0(A,B) ) ), file(binop_2,k11_binop_2), []). fof(redefinition_k12_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k12_binop_2(A,B) = k7_xcmplx_0(A,B) ) ), file(binop_2,k12_binop_2), []). fof(redefinition_k12_finseq_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => k12_finseq_1(A,B) = k5_finseq_1(B) ) ), file(finseq_1,k12_finseq_1), []). fof(redefinition_k14_integra1,axiom,( ! [A,B,C,D] : ( ( v1_integra1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & v1_funct_1(B) & m1_relset_1(B,A,k1_numbers) & ~ v1_xboole_0(C) & m2_integra1(C,A) & m1_subset_1(D,C) ) => k14_integra1(A,B,C,D) = k4_integra1(A,B,C,D) ) ), file(integra1,k14_integra1), []). fof(redefinition_k15_integra1,axiom,( ! [A,B,C,D] : ( ( v1_integra1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & v1_funct_1(B) & m1_relset_1(B,A,k1_numbers) & ~ v1_xboole_0(C) & m2_integra1(C,A) & m1_subset_1(D,C) ) => k15_integra1(A,B,C,D) = k5_integra1(A,B,C,D) ) ), file(integra1,k15_integra1), []). fof(redefinition_k1_goboard1,axiom,( ! [A,B] : ( ( m1_finseq_1(A,k1_numbers) & m1_subset_1(B,k5_numbers) ) => k1_goboard1(A,B) = k1_funct_1(A,B) ) ), file(goboard1,k1_goboard1), []). fof(redefinition_k1_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k1_nat_1(A,B) = k2_xcmplx_0(A,B) ) ), file(nat_1,k1_nat_1), []). fof(redefinition_k1_seq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) ) => k1_seq_1(A,B) = k1_funct_1(A,B) ) ), file(seq_1,k1_seq_1), []). fof(redefinition_k23_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k23_binop_2(A,B) = k2_xcmplx_0(A,B) ) ), file(binop_2,k23_binop_2), []). fof(redefinition_k2_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => k2_finseq_1(A) = k1_finseq_1(A) ) ), file(finseq_1,k2_finseq_1), []). fof(redefinition_k2_partfun1,axiom,( ! [A,B,C,D] : ( ( v1_funct_1(C) & m1_relset_1(C,A,B) ) => k2_partfun1(A,B,C,D) = k7_relat_1(C,D) ) ), file(partfun1,k2_partfun1), []). fof(redefinition_k2_seq_1,axiom,( ! [A,B,C,D] : ( ( v2_membered(B) & v1_funct_1(C) & m1_relset_1(C,A,B) ) => k2_seq_1(A,B,C,D) = k1_funct_1(C,D) ) ), file(seq_1,k2_seq_1), []). fof(redefinition_k3_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => k3_finseq_1(A) = k1_card_1(A) ) ), file(finseq_1,k3_finseq_1), []). fof(redefinition_k3_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => k3_pscomp_1(A) = k2_seq_4(A) ) ), file(pscomp_1,k3_pscomp_1), []). fof(redefinition_k4_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => k4_finseq_1(A) = k1_relat_1(A) ) ), file(finseq_1,k4_finseq_1), []). fof(redefinition_k4_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => k4_pscomp_1(A) = k3_seq_4(A) ) ), file(pscomp_1,k4_pscomp_1), []). fof(redefinition_k4_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => k4_relset_1(A,B,C) = k1_relat_1(C) ) ), file(relset_1,k4_relset_1), []). fof(redefinition_k5_numbers,axiom,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), []). fof(redefinition_k5_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k5_real_1(A,B) = k6_xcmplx_0(A,B) ) ), file(real_1,k5_real_1), []). fof(redefinition_k5_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => k5_relset_1(A,B,C) = k2_relat_1(C) ) ), file(relset_1,k5_relset_1), []). fof(redefinition_k5_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k5_subset_1(A,B,C) = k3_xboole_0(B,C) ) ), file(subset_1,k5_subset_1), []). fof(redefinition_k6_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k6_real_1(A,B) = k7_xcmplx_0(A,B) ) ), file(real_1,k6_real_1), []). fof(redefinition_k8_finseq_1,axiom,( ! [A,B,C] : ( ( m1_finseq_1(B,A) & m1_finseq_1(C,A) ) => k8_finseq_1(A,B,C) = k7_finseq_1(B,C) ) ), file(finseq_1,k8_finseq_1), []). fof(redefinition_k8_integra1,axiom,( ! [A] : ( ( v1_integra1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) ) => k8_integra1(A) = k1_integra1(A) ) ), file(integra1,k8_integra1), []). fof(redefinition_k9_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k9_binop_2(A,B) = k2_xcmplx_0(A,B) ) ), file(binop_2,k9_binop_2), []). fof(redefinition_k9_finseq_1,axiom,( ! [A] : k9_finseq_1(A) = k5_finseq_1(A) ), file(finseq_1,k9_finseq_1), []). fof(redefinition_m2_finseq_1,axiom,( ! [A,B] : ( m2_finseq_1(B,A) <=> m1_finseq_1(B,A) ) ), file(finseq_1,m2_finseq_1), []). fof(redefinition_m2_finseq_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & m1_finseq_2(B,A) ) => ! [C] : ( m2_finseq_2(C,A,B) <=> m1_subset_1(C,B) ) ) ), file(finseq_2,m2_finseq_2), []). fof(redefinition_m2_relset_1,axiom,( ! [A,B,C] : ( m2_relset_1(C,A,B) <=> m1_relset_1(C,A,B) ) ), file(relset_1,m2_relset_1), []). fof(redefinition_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,B) ) ) ), file(subset_1,m2_subset_1), []). fof(redefinition_m3_integra1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_rcomp_1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(B) & m2_integra1(B,A) ) => ! [C] : ( m3_integra1(C,A,B) <=> m1_subset_1(C,B) ) ) ), file(integra1,m3_integra1), []). fof(reflexivity_r1_tarski,axiom,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), []). fof(reflexivity_r1_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => r1_xreal_0(A,A) ) ), file(xreal_0,r1_xreal_0), []). fof(s1_finseq_2__e11_15__integra5,axiom,( ! [A,B,C,D] : ( ( v1_integra1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & v1_funct_1(B) & m2_relset_1(B,k1_numbers,k1_numbers) & m3_integra1(C,A,k8_integra1(A)) & m2_subset_1(D,k1_numbers,k5_numbers) ) => ? [E] : ( m2_finseq_1(E,k1_numbers) & k3_finseq_1(E) = D & ! [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) => ( r2_hidden(F,k2_finseq_1(D)) => k1_funct_1(E,F) = k2_seq_1(k1_numbers,k1_numbers,B,k3_pscomp_1(k2_integra1(A,k8_integra1(A),C,F))) ) ) ) ) ), file(integra5,s1_finseq_2__e11_15__integra5), []). fof(s1_finseq_2__e13_15__integra5,axiom,( ! [A,B,C,D] : ( ( v1_integra1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & v1_funct_1(B) & m2_relset_1(B,k1_numbers,k1_numbers) & m3_integra1(C,A,k8_integra1(A)) & m2_subset_1(D,k1_numbers,k5_numbers) ) => ? [E] : ( m2_finseq_1(E,k1_numbers) & k3_finseq_1(E) = D & ! [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) => ( r2_hidden(F,k2_finseq_1(D)) => k1_funct_1(E,F) = k2_seq_1(k1_numbers,k1_numbers,B,k4_pscomp_1(k2_integra1(A,k8_integra1(A),C,k23_binop_2(F,1)))) ) ) ) ) ), file(integra5,s1_finseq_2__e13_15__integra5), []). fof(s1_finseq_2__e7_15__integra5,axiom,( ! [A,B,C] : ( ( v1_integra1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & v1_funct_1(B) & m2_relset_1(B,k1_numbers,k1_numbers) & m3_integra1(C,A,k8_integra1(A)) ) => ? [D] : ( m2_finseq_1(D,k1_numbers) & k3_finseq_1(D) = k3_finseq_1(C) & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( r2_hidden(E,k2_finseq_1(k3_finseq_1(C))) => k1_funct_1(D,E) = k10_binop_2(k2_seq_1(k1_numbers,k1_numbers,B,k3_pscomp_1(k2_integra1(A,k8_integra1(A),C,E))),k2_seq_1(k1_numbers,k1_numbers,B,k4_pscomp_1(k2_integra1(A,k8_integra1(A),C,E)))) ) ) ) ) ), file(integra5,s1_finseq_2__e7_15__integra5), []). fof(spc0_boole,axiom,( v1_xboole_0(0) ), file(boole,spc0_boole), []). fof(spc1_boole,axiom,( ~ v1_xboole_0(1) ), file(boole,spc1_boole), []). fof(spc1_numerals,axiom, ( v2_xreal_0(1) & m1_subset_1(1,k5_numbers) ), file(numerals,spc1_numerals), []). fof(spc2_boole,axiom,( ~ v1_xboole_0(2) ), file(boole,spc2_boole), []). fof(spc2_numerals,axiom, ( v2_xreal_0(2) & m1_subset_1(2,k5_numbers) ), file(numerals,spc2_numerals), []). fof(t104_rvsum_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => ! [B] : ( m2_finseq_1(B,k1_numbers) => k15_rvsum_1(k8_finseq_1(k1_numbers,B,k12_finseq_1(k1_numbers,A))) = k9_binop_2(k15_rvsum_1(B),A) ) ) ), file(rvsum_1,t104_rvsum_1), []). fof(t106_rvsum_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => ! [B] : ( m2_finseq_1(B,k1_numbers) => k15_rvsum_1(k8_finseq_1(k1_numbers,k12_finseq_1(k1_numbers,A),B)) = k9_binop_2(A,k15_rvsum_1(B)) ) ) ), file(rvsum_1,t106_rvsum_1), []). fof(t10_finseq_2,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v1_finseq_1(C) ) => ( ( k3_finseq_1(B) = A & k3_finseq_1(C) = A & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( r2_hidden(D,k2_finseq_1(A)) => k1_funct_1(B,D) = k1_funct_1(C,D) ) ) ) => B = C ) ) ) ) ), file(finseq_2,t10_finseq_2), []). fof(t10_integra1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( v1_integra1(B) & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) ) => ! [C] : ( ( ~ v1_xboole_0(C) & m2_integra1(C,B) ) => ! [D] : ( m3_integra1(D,B,C) => ( r2_hidden(A,k4_finseq_1(D)) => r1_tarski(k2_integra1(B,C,D,A),B) ) ) ) ) ) ), file(integra1,t10_integra1), []). fof(t110_finseq_2,axiom,( ! [A,B] : ( m2_finseq_1(B,A) => m1_subset_1(B,k4_finseq_2(k3_finseq_1(B),A)) ) ), file(finseq_2,t110_finseq_2), []). fof(t112_rvsum_1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers)) => ! [C] : ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers)) => ( ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( r2_hidden(D,k2_finseq_1(A)) => r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,B,D),k2_seq_1(k5_numbers,k1_numbers,C,D)) ) ) => r1_xreal_0(k15_rvsum_1(B),k15_rvsum_1(C)) ) ) ) ) ), file(rvsum_1,t112_rvsum_1), []). fof(t11_integra1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v3_seq_4(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) ) => r1_xreal_0(0,k3_integra1(A)) ) ), file(integra1,t11_integra1), []). fof(t11_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( r1_xreal_0(A,B) <=> r1_xreal_0(k6_xcmplx_0(A,C),k6_xcmplx_0(B,C)) ) ) ) ) ), file(xreal_1,t11_xreal_1), []). fof(t15_rcomp_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ~ ( ~ r1_xreal_0(B,A) & k2_rcomp_1(A,B) = k1_xboole_0 ) & ( r1_xreal_0(A,B) => ( r2_hidden(A,k1_rcomp_1(A,B)) & r2_hidden(B,k1_rcomp_1(A,B)) ) ) & r1_tarski(k2_rcomp_1(A,B),k1_rcomp_1(A,B)) ) ) ) ), file(rcomp_1,t15_rcomp_1), []). fof(t16_xboole_1,axiom,( ! [A,B,C] : k3_xboole_0(k3_xboole_0(A,B),C) = k3_xboole_0(A,k3_xboole_0(B,C)) ), file(xboole_1,t16_xboole_1), []). fof(t17_fcont_1,axiom,( ! [A,B,C] : ( ( v1_funct_1(C) & m2_relset_1(C,k1_numbers,k1_numbers) ) => ( ( r2_fcont_1(C,A) & r1_tarski(B,A) ) => r2_fcont_1(C,B) ) ) ), file(fcont_1,t17_fcont_1), []). fof(t18_integra4,axiom,( ! [A] : ( ( v1_integra1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) ) => ! [B,C] : ( ( v1_funct_1(C) & m2_relset_1(C,A,k1_numbers) ) => ( r1_rfunct_1(C,A) => v1_seq_4(k5_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,C,B))) ) ) ) ), file(integra4,t18_integra4), []). fof(t19_integra4,axiom,( ! [A] : ( ( v1_integra1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) ) => ! [B,C] : ( ( v1_funct_1(C) & m2_relset_1(C,A,k1_numbers) ) => ( r2_rfunct_1(C,A) => v2_seq_4(k5_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,C,B))) ) ) ) ), file(integra4,t19_integra4), []). fof(t1_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k2_xcmplx_0(A,0) = A ) ), file(arithm,t1_arithm), []). fof(t1_integra2,axiom,( ! [A] : ( ( v1_integra1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) ) => ! [B] : ( v1_xreal_0(B) => ( r2_hidden(B,A) <=> ( r1_xreal_0(k4_pscomp_1(A),B) & r1_xreal_0(B,k3_pscomp_1(A)) ) ) ) ) ), file(integra2,t1_integra2), []). fof(t1_numerals,axiom,( m1_subset_1(0,k5_numbers) ), file(numerals,t1_numerals), []). fof(t1_real,axiom,( ! [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), []). fof(t1_subset,axiom,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), []). fof(t1_xboole_1,axiom,( ! [A,B,C] : ( ( r1_tarski(A,B) & r1_tarski(B,C) ) => r1_tarski(A,C) ) ), file(xboole_1,t1_xboole_1), []). fof(t21_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( r1_xreal_0(k2_xcmplx_0(A,B),C) <=> r1_xreal_0(A,k6_xcmplx_0(C,B)) ) ) ) ) ), file(xreal_1,t21_xreal_1), []). fof(t22_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( r1_xreal_0(A,k2_xcmplx_0(B,C)) <=> r1_xreal_0(k6_xcmplx_0(A,B),C) ) ) ) ) ), file(xreal_1,t22_xreal_1), []). fof(t24_integra1,axiom,( ! [A] : ( m2_finseq_1(A,k1_numbers) => ! [B] : ( m2_finseq_1(B,k1_numbers) => ! [C] : ( m2_finseq_1(C,k1_numbers) => ( ( k3_finseq_1(A) = k3_finseq_1(B) & k3_finseq_1(A) = k3_finseq_1(C) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( r2_hidden(D,k4_finseq_1(A)) => k1_goboard1(C,D) = k10_binop_2(k4_finseq_4(k5_numbers,k1_numbers,A,D),k4_finseq_4(k5_numbers,k1_numbers,B,D)) ) ) ) => k15_rvsum_1(C) = k10_binop_2(k15_rvsum_1(A),k15_rvsum_1(B)) ) ) ) ) ), file(integra1,t24_integra1), []). fof(t25_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( k3_finseq_1(A) = 0 <=> A = k1_xboole_0 ) ) ), file(finseq_1,t25_finseq_1), []). fof(t28_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ~ ( r1_xreal_0(A,B) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => B != k2_xcmplx_0(A,C) ) ) ) ) ), file(nat_1,t28_nat_1), []). fof(t28_partfun1,axiom,( ! [A,B,C,D] : ( ( v1_funct_1(D) & m2_relset_1(D,A,B) ) => ( r1_tarski(k4_relset_1(A,B,D),C) => ( v1_funct_1(D) & m2_relset_1(D,C,B) ) ) ) ), file(partfun1,t28_partfun1), []). fof(t28_xboole_1,axiom,( ! [A,B] : ( r1_tarski(A,B) => k3_xboole_0(A,B) = A ) ), file(xboole_1,t28_xboole_1), []). fof(t29_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => r1_xreal_0(A,k2_xcmplx_0(A,B)) ) ) ), file(nat_1,t29_nat_1), []). fof(t2_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(A,0) = 0 ) ), file(arithm,t2_arithm), []). fof(t2_boole,axiom,( ! [A] : k3_xboole_0(A,k1_xboole_0) = k1_xboole_0 ), file(boole,t2_boole), []). fof(t2_real,axiom,( ! [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), []). fof(t2_subset,axiom,( ! [A,B] : ( m1_subset_1(A,B) => ( v1_xboole_0(B) | r2_hidden(A,B) ) ) ), file(subset,t2_subset), []). fof(t31_nat_2,axiom,( ! [A] : ( ( v4_ordinal2(A) & ~ v1_realset1(A) ) => r1_xreal_0(2,A) ) ), file(nat_2,t31_nat_2), []). fof(t33_fdiff_1,axiom,( ! [A,B] : ( ( v1_funct_1(B) & m2_relset_1(B,k1_numbers,k1_numbers) ) => ( r2_fdiff_1(B,A) => r2_fcont_1(B,A) ) ) ), file(fdiff_1,t33_fdiff_1), []). fof(t34_fdiff_1,axiom,( ! [A,B] : ( ( v3_rcomp_1(B) & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) ) => ! [C] : ( ( v1_funct_1(C) & m2_relset_1(C,k1_numbers,k1_numbers) ) => ( ( r2_fdiff_1(C,A) & r1_tarski(B,A) ) => r2_fdiff_1(C,B) ) ) ) ), file(fdiff_1,t34_fdiff_1), []). fof(t35_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => k3_finseq_1(k7_finseq_1(A,B)) = k1_nat_1(k3_finseq_1(A),k3_finseq_1(B)) ) ) ), file(finseq_1,t35_finseq_1), []). fof(t36_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ! [C] : ( v4_ordinal2(C) => ( ( r1_xreal_0(k1_nat_1(k3_finseq_1(A),1),C) & r1_xreal_0(C,k1_nat_1(k3_finseq_1(A),k3_finseq_1(B))) ) => k1_funct_1(k7_finseq_1(A,B),C) = k1_funct_1(B,k6_xcmplx_0(C,k3_finseq_1(A))) ) ) ) ) ), file(finseq_1,t36_finseq_1), []). fof(t38_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( ~ r1_xreal_0(k2_xcmplx_0(B,1),A) <=> r1_xreal_0(A,B) ) ) ) ), file(nat_1,t38_nat_1), []). fof(t39_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ( ~ r1_xreal_0(1,A) => A = 0 ) ) ), file(nat_1,t39_nat_1), []). fof(t3_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(1,A) = A ) ), file(arithm,t3_arithm), []). fof(t3_finseq_1,axiom,( ! [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), []). fof(t3_real,axiom,( ! [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), []). fof(t3_rolle,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ( ~ r1_xreal_0(B,A) => ! [C] : ( ( v1_funct_1(C) & m2_relset_1(C,k1_numbers,k1_numbers) ) => ~ ( r2_fcont_1(C,k1_rcomp_1(A,B)) & r2_fdiff_1(C,k2_rcomp_1(A,B)) & ! [D] : ( m1_subset_1(D,k1_numbers) => ~ ( r2_hidden(D,k2_rcomp_1(A,B)) & k1_fdiff_1(C,D) = k6_real_1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,C,B),k2_seq_1(k1_numbers,k1_numbers,C,A)),k5_real_1(B,A)) ) ) ) ) ) ) ) ), file(rolle,t3_rolle), []). fof(t3_subset,axiom,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), []). fof(t47_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( k7_finseq_1(A,k1_xboole_0) = A & k7_finseq_1(k1_xboole_0,A) = A ) ) ), file(finseq_1,t47_finseq_1), []). fof(t4_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k6_xcmplx_0(A,0) = A ) ), file(arithm,t4_arithm), []). fof(t4_finseq_1,axiom, ( 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), []). fof(t4_integra1,axiom,( ! [A] : ( ( v1_integra1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) ) => ? [B] : ( m1_subset_1(B,k1_numbers) & ? [C] : ( m1_subset_1(C,k1_numbers) & r1_xreal_0(B,C) & B = k4_pscomp_1(A) & C = k3_pscomp_1(A) ) ) ) ), file(integra1,t4_integra1), []). fof(t4_real,axiom,( ! [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), []). fof(t4_subset,axiom,( ! [A,B,C] : ( ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) ) => m1_subset_1(A,C) ) ), file(subset,t4_subset), []). fof(t52_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(B,A) & r1_xreal_0(k6_xcmplx_0(B,A),0) ) ) ) ), file(xreal_1,t52_xreal_1), []). fof(t56_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( B = k9_finseq_1(A) <=> ( k3_finseq_1(B) = 1 & k2_relat_1(B) = k1_tarski(A) ) ) ) ), file(finseq_1,t56_finseq_1), []). fof(t57_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( B = k9_finseq_1(A) <=> ( k3_finseq_1(B) = 1 & k1_funct_1(B,1) = A ) ) ) ), file(finseq_1,t57_finseq_1), []). fof(t5_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k7_xcmplx_0(0,A) = 0 ) ), file(arithm,t5_arithm), []). fof(t5_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => ( A = 0 | r2_hidden(A,k2_finseq_1(A)) ) ) ), file(finseq_1,t5_finseq_1), []). fof(t5_integra1,axiom,( ! [A] : ( ( v1_integra1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) ) => A = k1_rcomp_1(k4_pscomp_1(A),k3_pscomp_1(A)) ) ), file(integra1,t5_integra1), []). fof(t5_real,axiom,( ! [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), []). fof(t5_subset,axiom,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), []). fof(t66_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ( r1_xreal_0(A,B) & r1_xreal_0(0,C) ) => r1_xreal_0(k3_xcmplx_0(A,C),k3_xcmplx_0(B,C)) ) ) ) ) ), file(xreal_1,t66_xreal_1), []). fof(t68_funct_1,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( B = k7_relat_1(C,A) <=> ( k1_relat_1(B) = k3_xboole_0(k1_relat_1(C),A) & ! [D] : ( r2_hidden(D,k1_relat_1(B)) => k1_funct_1(B,D) = k1_funct_1(C,D) ) ) ) ) ) ), file(funct_1,t68_funct_1), []). fof(t6_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k7_xcmplx_0(A,1) = A ) ), file(arithm,t6_arithm), []). fof(t6_boole,axiom,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), []). fof(t6_real,axiom,( ! [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), []). fof(t7_boole,axiom,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), []). fof(t7_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( r1_xreal_0(A,B) <=> r1_tarski(k2_finseq_1(A),k2_finseq_1(B)) ) ) ) ), file(finseq_1,t7_finseq_1), []). fof(t7_real,axiom,( ! [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), []). fof(t88_xcmplx_1,axiom,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => ( A != 0 => k3_xcmplx_0(k7_xcmplx_0(B,A),A) = B ) ) ) ), file(xcmplx_1,t88_xcmplx_1), []). fof(t8_boole,axiom,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), []). fof(t8_real,axiom,( ! [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), []). fof(t8_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( r1_xreal_0(A,B) <=> r1_xreal_0(k2_xcmplx_0(A,C),k2_xcmplx_0(B,C)) ) ) ) ) ), file(xreal_1,t8_xreal_1), []). fof(t9_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ( ( r1_xreal_0(A,B) & r1_xreal_0(C,D) ) => r1_xreal_0(k2_xcmplx_0(A,C),k2_xcmplx_0(B,D)) ) ) ) ) ) ), file(xreal_1,t9_xreal_1), []).