% Mizar problem: t61_nat_3,nat_3,2298,12 fof(t61_nat_3,conjecture,( ! [A] : ( ( ~ v1_xboole_0(A) & v4_ordinal2(A) ) => k8_nat_3(k12_newton,k12_nat_3(A)) = A ) ), inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc10_circcmb3,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_card_5,cc1_finset_1,cc1_funct_1,cc1_membered,cc1_nat_1,cc1_nat_3,cc1_polynom1,cc1_relat_1,cc1_relset_1,cc1_seq_1,cc1_tex_2,cc1_xcmplx_0,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_funct_1,cc2_int_1,cc2_membered,cc2_nat_1,cc2_nat_3,cc2_tex_2,cc2_xcmplx_0,cc2_xreal_0,cc3_int_1,cc3_membered,cc3_nat_1,cc3_polynom1,cc3_tex_2,cc3_xreal_0,cc4_int_1,cc4_membered,cc4_tex_2,cc4_xreal_0,cc5_polynom1,cc5_tex_2,cc5_xreal_0,cc6_membered,cc6_polynom1,cc6_tex_2,cc6_xreal_0,cc7_tex_2,cc7_xreal_0,cc8_circcmb3,cc8_xreal_0,cc9_circcmb3,cc9_membered,commutativity_k23_binop_2,commutativity_k24_binop_2,commutativity_k2_tarski,commutativity_k2_xcmplx_0,commutativity_k3_xboole_0,commutativity_k3_xcmplx_0,commutativity_k5_subset_1,commutativity_k6_nat_1,commutativity_k9_polynom1,connectedness_r1_xreal_0,d10_xboole_0,d15_polynom1,d1_finseq_1,d1_pre_circ,d1_tarski,d1_xboole_0,d21_ordinal2,d3_nat_1,d3_pboole,d3_tarski,d3_xboole_0,d5_finseq_1,d5_int_2,d5_nat_1,d5_nat_3,d5_real_1,d5_tarski,d6_int_2,d7_nat_3,d7_polynom1,d7_xboole_0,d8_nat_3,d9_nat_3,dt_k10_nat_3,dt_k10_wsierp_1,dt_k11_nat_3,dt_k11_polynom1,dt_k12_finseq_1,dt_k12_nat_3,dt_k12_newton,dt_k13_binarith,dt_k13_polynom1,dt_k14_polynom1,dt_k16_polynom1,dt_k16_rvsum_1,dt_k1_finseq_1,dt_k1_funct_1,dt_k1_numbers,dt_k1_polynom2,dt_k1_pre_circ,dt_k1_relat_1,dt_k1_tarski,dt_k1_uproots,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k23_binop_2,dt_k24_binop_2,dt_k2_finseq_1,dt_k2_funcop_1,dt_k2_newton,dt_k2_seq_4,dt_k2_tarski,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k3_newton,dt_k3_xboole_0,dt_k3_xcmplx_0,dt_k4_finseq_2,dt_k4_tarski,dt_k5_facirc_1,dt_k5_finseq_1,dt_k5_numbers,dt_k5_ordinal2,dt_k5_relat_1,dt_k5_subset_1,dt_k6_nat_1,dt_k7_nat_3,dt_k8_nat_3,dt_k8_polynom1,dt_k9_finseq_1,dt_k9_polynom1,dt_m1_circcomb,dt_m1_finseq_1,dt_m1_finseq_2,dt_m1_pboole,dt_m1_polynom1,dt_m1_relset_1,dt_m1_subset_1,dt_m1_trees_4,dt_m2_finseq_1,dt_m2_finseq_2,dt_m2_relset_1,dt_m2_subset_1,existence_m1_circcomb,existence_m1_finseq_1,existence_m1_finseq_2,existence_m1_pboole,existence_m1_polynom1,existence_m1_relset_1,existence_m1_subset_1,existence_m1_trees_4,existence_m2_finseq_1,existence_m2_finseq_2,existence_m2_relset_1,existence_m2_subset_1,fc10_finset_1,fc10_membered,fc10_relat_1,fc10_xreal_0,fc11_finset_1,fc11_membered,fc11_polynom1,fc11_xreal_0,fc12_membered,fc12_relat_1,fc12_xreal_0,fc13_membered,fc14_finset_1,fc14_membered,fc14_polynom1,fc15_membered,fc16_membered,fc17_nat_3,fc17_polynom1,fc18_nat_3,fc18_polynom1,fc19_nat_3,fc1_finset_1,fc1_funct_1,fc1_int_1,fc1_nat_1,fc1_newton,fc1_pre_circ,fc1_relat_1,fc1_seq_1,fc1_tex_2,fc20_nat_3,fc20_polynom1,fc21_nat_3,fc21_polynom1,fc21_xreal_0,fc22_nat_3,fc22_xreal_0,fc23_polynom1,fc23_xreal_0,fc24_xreal_0,fc25_polynom1,fc27_membered,fc28_membered,fc29_membered,fc2_finset_1,fc2_int_1,fc2_membered,fc2_nat_1,fc2_nat_3,fc2_newton,fc2_xcmplx_0,fc30_membered,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc3_nat_1,fc3_polynom1,fc3_xcmplx_0,fc3_xreal_0,fc4_nat_1,fc4_relat_1,fc4_xreal_0,fc5_membered,fc5_newton,fc5_polynom1,fc5_relat_1,fc6_int_1,fc6_membered,fc7_circcmb3,fc7_int_1,fc7_membered,fc7_relat_1,fc7_xreal_0,fc8_membered,fc8_xcmplx_0,fc8_xreal_0,fc9_membered,fc9_relat_1,fc9_xreal_0,fraenkel_a_1_0_finseq_1,idempotence_k3_xboole_0,idempotence_k5_subset_1,idempotence_k6_nat_1,rc10_polynom1,rc11_polynom1,rc13_polynom1,rc1_card_5,rc1_finset_1,rc1_funct_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_nat_3,rc1_newton,rc1_pre_circ,rc1_relat_1,rc1_seq_1,rc1_tex_2,rc1_xcmplx_0,rc1_xreal_0,rc2_finset_1,rc2_funct_1,rc2_int_1,rc2_nat_1,rc2_nat_3,rc2_polynom1,rc2_relat_1,rc2_tex_2,rc2_xcmplx_0,rc2_xreal_0,rc3_finset_1,rc3_funct_1,rc3_nat_1,rc3_polynom1,rc3_relat_1,rc3_tex_2,rc3_xreal_0,rc4_finset_1,rc4_funct_1,rc4_polynom1,rc4_tex_2,rc4_xreal_0,rc5_tex_2,rc6_circcmb3,rc6_tex_2,rc7_circcmb3,rc7_polynom1,rc7_tex_2,rc8_polynom1,rc9_polynom1,redefinition_k10_wsierp_1,redefinition_k12_finseq_1,redefinition_k13_binarith,redefinition_k14_polynom1,redefinition_k1_polynom2,redefinition_k1_pre_circ,redefinition_k23_binop_2,redefinition_k24_binop_2,redefinition_k2_finseq_1,redefinition_k3_newton,redefinition_k5_facirc_1,redefinition_k5_numbers,redefinition_k5_subset_1,redefinition_k8_nat_3,redefinition_k8_polynom1,redefinition_k9_finseq_1,redefinition_m1_polynom1,redefinition_m1_trees_4,redefinition_m2_finseq_1,redefinition_m2_finseq_2,redefinition_m2_relset_1,redefinition_m2_subset_1,redefinition_r6_pboole,reflexivity_r1_nat_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,reflexivity_r6_pboole,s2_nat_1__e3_101__nat_3,spc0_boole,spc1_boole,spc1_numerals,spc2_boole,spc2_numerals,symmetry_r1_xboole_0,symmetry_r2_int_2,symmetry_r6_pboole,t11_newton,t125_rvsum_1,t19_nat_3,t1_arithm,t1_numerals,t1_real,t1_subset,t1_uproots,t1_xboole_1,t20_bagorder,t20_nat_3,t28_nat_1,t2_arithm,t2_boole,t2_real,t2_subset,t2_tarski,t2_xboole_1,t34_nat_3,t36_circcmb3,t37_nat_3,t38_nat_1,t3_arithm,t3_finseq_1,t3_real,t3_subset,t42_nat_3,t45_nat_3,t48_int_2,t4_finseq_1,t4_real,t4_subset,t51_nat_1,t58_nat_3,t59_nat_3,t5_int_2,t5_nat_3,t5_real,t5_subset,t6_boole,t6_real,t6_uproots,t6_xcmplx_1,t7_boole,t7_real,t8_boole,t8_real]), [file(nat_3,t61_nat_3)]). fof(antisymmetry_r2_hidden,axiom,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), []). fof(cc10_circcmb3,axiom,( ! [A] : ( m1_finseq_1(A,k5_numbers) => v7_seqm_3(A) ) ), file(circcmb3,cc10_circcmb3), []). 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_card_5,axiom,( ! [A] : ( ~ v1_finset_1(A) => ~ v1_xboole_0(A) ) ), file(card_5,cc1_card_5), []). fof(cc1_finset_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_finset_1(A) ) ), file(finset_1,cc1_finset_1), []). fof(cc1_funct_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_funct_1(A) ) ), file(funct_1,cc1_funct_1), []). fof(cc1_membered,axiom,( ! [A] : ( v5_membered(A) => v4_membered(A) ) ), file(membered,cc1_membered), []). fof(cc1_nat_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(nat_1,cc1_nat_1), []). fof(cc1_nat_3,axiom,( ! [A] : ( ( v1_relat_1(A) & v7_seqm_3(A) ) => ( v1_relat_1(A) & v1_seq_1(A) ) ) ), file(nat_3,cc1_nat_3), []). fof(cc1_polynom1,axiom,( ! [A] : ( m1_finseq_1(A,k5_numbers) => v1_card_3(A) ) ), file(polynom1,cc1_polynom1), []). fof(cc1_relat_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_relat_1(A) ) ), file(relat_1,cc1_relat_1), []). 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_tex_2,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( ~ v1_tex_2(B,k1_zfmisc_1(A)) => ~ v1_xboole_0(B) ) ) ) ), file(tex_2,cc1_tex_2), []). fof(cc1_xcmplx_0,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => v1_xcmplx_0(A) ) ), file(xcmplx_0,cc1_xcmplx_0), []). 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_finset_1,axiom,( ! [A] : ( v1_finset_1(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_finset_1(B) ) ) ), file(finset_1,cc2_finset_1), []). fof(cc2_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ) ), file(funct_1,cc2_funct_1), []). fof(cc2_int_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_int_1(A) ) ) ), file(int_1,cc2_int_1), []). fof(cc2_membered,axiom,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), []). fof(cc2_nat_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc2_nat_1), []). fof(cc2_nat_3,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_int_2(A) => ( ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xreal_0(A) & v2_xreal_0(A) & ~ v3_xreal_0(A) & v1_int_1(A) & v1_xcmplx_0(A) & v1_int_2(A) ) ) ) ), file(nat_3,cc2_nat_3), []). fof(cc2_tex_2,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_xboole_0(B) => v1_tex_2(B,k1_zfmisc_1(A)) ) ) ) ), file(tex_2,cc2_tex_2), []). fof(cc2_xcmplx_0,axiom,( ! [A] : ( v4_ordinal2(A) => v1_xcmplx_0(A) ) ), file(xcmplx_0,cc2_xcmplx_0), []). fof(cc2_xreal_0,axiom,( ! [A] : ( v1_xreal_0(A) => v1_xcmplx_0(A) ) ), file(xreal_0,cc2_xreal_0), []). fof(cc3_int_1,axiom,( ! [A] : ( v4_ordinal2(A) => v1_int_1(A) ) ), file(int_1,cc3_int_1), []). fof(cc3_membered,axiom,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), []). fof(cc3_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ( v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc3_nat_1), []). fof(cc3_polynom1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v7_seqm_3(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) ) ) ), file(polynom1,cc3_polynom1), []). fof(cc3_tex_2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_realset1(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_tex_2(B,k1_zfmisc_1(A)) => ( v1_xboole_0(B) & v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) & v5_membered(B) & v1_tex_2(B,k1_zfmisc_1(A)) ) ) ) ) ), file(tex_2,cc3_tex_2), []). 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_int_1,axiom,( ! [A] : ( v1_int_1(A) => ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(int_1,cc4_int_1), []). fof(cc4_membered,axiom,( ! [A] : ( v2_membered(A) => v1_membered(A) ) ), file(membered,cc4_membered), []). fof(cc4_tex_2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_realset1(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( ~ v1_xboole_0(B) => ( ~ v1_xboole_0(B) & ~ v1_tex_2(B,k1_zfmisc_1(A)) ) ) ) ) ), file(tex_2,cc4_tex_2), []). 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_polynom1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v1_polynom1(A) ) ) ), file(polynom1,cc5_polynom1), []). fof(cc5_tex_2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_realset1(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( ( ~ v1_xboole_0(B) & ~ v1_tex_2(B,k1_zfmisc_1(A)) ) => ( ~ v1_xboole_0(B) & v1_realset1(B) & ~ v1_tex_2(B,k1_zfmisc_1(A)) ) ) ) ) ), file(tex_2,cc5_tex_2), []). 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_polynom1,axiom,( ! [A] : ( v1_finset_1(A) => ! [B] : ( m1_pboole(B,A) => v1_polynom1(B) ) ) ), file(polynom1,cc6_polynom1), []). fof(cc6_tex_2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & ~ v1_realset1(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( ( ~ v1_xboole_0(B) & v1_realset1(B) ) => ( ~ v1_xboole_0(B) & v1_tex_2(B,k1_zfmisc_1(A)) ) ) ) ) ), file(tex_2,cc6_tex_2), []). 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_tex_2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & ~ v1_realset1(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( ( ~ v1_xboole_0(B) & ~ v1_tex_2(B,k1_zfmisc_1(A)) ) => ( ~ v1_xboole_0(B) & ~ v1_realset1(B) ) ) ) ) ), file(tex_2,cc7_tex_2), []). 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_circcmb3,axiom,( ! [A] : ( ~ v2_facirc_1(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ~ v2_facirc_1(B) ) ) ), file(circcmb3,cc8_circcmb3), []). 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_circcmb3,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v7_seqm_3(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v3_facirc_1(A) ) ) ), file(circcmb3,cc9_circcmb3), []). 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_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_k24_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k24_binop_2(A,B) = k24_binop_2(B,A) ) ), file(binop_2,k24_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_k6_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k6_nat_1(A,B) = k6_nat_1(B,A) ) ), file(nat_1,k6_nat_1), []). fof(commutativity_k9_polynom1,axiom,( ! [A,B,C] : ( ( v1_seq_1(B) & m1_pboole(B,A) & v1_seq_1(C) & m1_pboole(C,A) ) => k9_polynom1(A,B,C) = k9_polynom1(A,C,B) ) ), file(polynom1,k9_polynom1), []). 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_xboole_0,axiom,( ! [A,B] : ( A = B <=> ( r1_tarski(A,B) & r1_tarski(B,A) ) ) ), file(xboole_0,d10_xboole_0), []). fof(d15_polynom1,axiom,( ! [A] : k16_polynom1(A) = k2_funcop_1(A,0) ), file(polynom1,d15_polynom1), []). fof(d1_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => k1_finseq_1(A) = a_1_0_finseq_1(A) ) ), file(finseq_1,d1_finseq_1), []). fof(d1_pre_circ,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_finset_1(A) & v2_membered(A) ) => ! [B] : ( v1_xreal_0(B) => ( B = k1_pre_circ(A) <=> ( r2_hidden(B,A) & ! [C] : ( v1_xreal_0(C) => ( r2_hidden(C,A) => r1_xreal_0(C,B) ) ) ) ) ) ) ), file(pre_circ,d1_pre_circ), []). fof(d1_tarski,axiom,( ! [A,B] : ( B = k1_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> C = A ) ) ), file(tarski,d1_tarski), []). fof(d1_xboole_0,axiom,( ! [A] : ( A = k1_xboole_0 <=> ! [B] : ~ r2_hidden(B,A) ) ), file(xboole_0,d1_xboole_0), []). fof(d21_ordinal2,axiom,( ! [A] : ( v4_ordinal2(A) <=> r2_hidden(A,k5_ordinal2) ) ), file(ordinal2,d21_ordinal2), []). fof(d3_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( r1_nat_1(A,B) <=> ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & B = k3_xcmplx_0(A,C) ) ) ) ) ), file(nat_1,d3_nat_1), []). fof(d3_pboole,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( m1_pboole(B,A) <=> k1_relat_1(B) = A ) ) ), file(pboole,d3_pboole), []). fof(d3_tarski,axiom,( ! [A,B] : ( r1_tarski(A,B) <=> ! [C] : ( r2_hidden(C,A) => r2_hidden(C,B) ) ) ), file(tarski,d3_tarski), []). fof(d3_xboole_0,axiom,( ! [A,B,C] : ( C = k3_xboole_0(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( r2_hidden(D,A) & r2_hidden(D,B) ) ) ) ), file(xboole_0,d3_xboole_0), []). fof(d5_finseq_1,axiom,( ! [A] : k5_finseq_1(A) = k1_tarski(k4_tarski(1,A)) ), file(finseq_1,d5_finseq_1), []). fof(d5_int_2,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( v1_int_2(A) <=> ( ~ r1_xreal_0(A,1) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ ( r1_nat_1(B,A) & B != 1 & B != A ) ) ) ) ) ), file(int_2,d5_int_2), []). fof(d5_nat_1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( C = k6_nat_1(A,B) <=> ( r1_nat_1(C,A) & r1_nat_1(C,B) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( ( r1_nat_1(D,A) & r1_nat_1(D,B) ) => r1_nat_1(D,C) ) ) ) ) ) ) ) ), file(nat_1,d5_nat_1), []). fof(d5_nat_3,axiom,( ! [A,B] : ( ( v7_seqm_3(B) & v1_polynom1(B) & m1_pboole(B,A) ) => ! [C] : ( v4_ordinal2(C) => ( C = k7_nat_3(A,B) <=> ? [D] : ( m1_trees_4(D,k1_numbers,k5_numbers) & C = k10_wsierp_1(D) & D = k5_relat_1(k1_uproots(k1_polynom2(A,B)),B) ) ) ) ) ), file(nat_3,d5_nat_3), []). 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_tarski,axiom,( ! [A,B] : k4_tarski(A,B) = k2_tarski(k2_tarski(A,B),k1_tarski(A)) ), file(tarski,d5_tarski), []). fof(d6_int_2,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( r2_int_2(A,B) <=> k6_nat_1(A,B) = 1 ) ) ) ), file(int_2,d6_int_2), []). fof(d7_nat_3,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ~ ( B != 1 & A != 0 & ~ ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( C = k10_nat_3(A,B) <=> ( r1_nat_1(k2_newton(B,C),A) & ~ r1_nat_1(k2_newton(B,k23_binop_2(C,1)),A) ) ) ) ) ) ) ), file(nat_3,d7_nat_3), []). fof(d7_polynom1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( B = k11_polynom1(A) <=> ! [C] : ( r2_hidden(C,B) <=> k1_funct_1(A,C) != 0 ) ) ) ), file(polynom1,d7_polynom1), []). fof(d7_xboole_0,axiom,( ! [A,B] : ( r1_xboole_0(A,B) <=> k3_xboole_0(A,B) = k1_xboole_0 ) ), file(xboole_0,d7_xboole_0), []). fof(d8_nat_3,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( m1_pboole(B,k12_newton) => ( B = k11_nat_3(A) <=> ! [C] : ( ( v1_int_2(C) & m2_subset_1(C,k1_numbers,k5_numbers) ) => k1_funct_1(B,C) = k10_nat_3(A,C) ) ) ) ) ), file(nat_3,d8_nat_3), []). fof(d9_nat_3,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v4_ordinal2(A) ) => ! [B] : ( m1_pboole(B,k12_newton) => ( B = k12_nat_3(A) <=> ( k11_polynom1(B) = k1_polynom2(k12_newton,k11_nat_3(A)) & ! [C] : ( v4_ordinal2(C) => ( r2_hidden(C,k1_polynom2(k12_newton,k11_nat_3(A))) => k1_funct_1(B,C) = k2_newton(C,k10_nat_3(A,C)) ) ) ) ) ) ) ), file(nat_3,d9_nat_3), []). fof(dt_k10_nat_3,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => m2_subset_1(k10_nat_3(A,B),k1_numbers,k5_numbers) ) ), file(nat_3,k10_nat_3), []). fof(dt_k10_wsierp_1,axiom,( ! [A] : ( m1_finseq_1(A,k5_numbers) => m2_subset_1(k10_wsierp_1(A),k1_numbers,k5_numbers) ) ), file(wsierp_1,k10_wsierp_1), []). fof(dt_k11_nat_3,axiom,( ! [A] : ( v4_ordinal2(A) => m1_pboole(k11_nat_3(A),k12_newton) ) ), file(nat_3,k11_nat_3), []). fof(dt_k11_polynom1,axiom,( $true ), file(polynom1,k11_polynom1), []). 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_k12_nat_3,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v4_ordinal2(A) ) => m1_pboole(k12_nat_3(A),k12_newton) ) ), file(nat_3,k12_nat_3), []). fof(dt_k12_newton,axiom,( m1_subset_1(k12_newton,k1_zfmisc_1(k5_numbers)) ), file(newton,k12_newton), []). fof(dt_k13_binarith,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => m2_finseq_2(k13_binarith(A,B),A,k4_finseq_2(1,A)) ) ), file(binarith,k13_binarith), []). fof(dt_k13_polynom1,axiom,( $true ), file(polynom1,k13_polynom1), []). fof(dt_k14_polynom1,axiom,( ! [A] : m1_subset_1(k14_polynom1(A),k1_zfmisc_1(k13_polynom1(A))) ), file(polynom1,k14_polynom1), []). fof(dt_k16_polynom1,axiom,( ! [A] : m1_polynom1(k16_polynom1(A),A,k14_polynom1(A)) ), file(polynom1,k16_polynom1), []). fof(dt_k16_rvsum_1,axiom,( ! [A] : ( m1_finseq_1(A,k1_numbers) => m1_subset_1(k16_rvsum_1(A),k1_numbers) ) ), file(rvsum_1,k16_rvsum_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_numbers,axiom,( $true ), file(numbers,k1_numbers), []). fof(dt_k1_polynom2,axiom,( ! [A,B] : ( ( v7_seqm_3(B) & v1_polynom1(B) & m1_pboole(B,A) ) => ( v1_finset_1(k1_polynom2(A,B)) & m1_subset_1(k1_polynom2(A,B),k1_zfmisc_1(A)) ) ) ), file(polynom2,k1_polynom2), []). fof(dt_k1_pre_circ,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_finset_1(A) & v2_membered(A) ) => v1_xreal_0(k1_pre_circ(A)) ) ), file(pre_circ,k1_pre_circ), []). fof(dt_k1_relat_1,axiom,( $true ), file(relat_1,k1_relat_1), []). fof(dt_k1_tarski,axiom,( $true ), file(tarski,k1_tarski), []). fof(dt_k1_uproots,axiom,( ! [A] : ( v1_finset_1(A) => m2_finseq_1(k1_uproots(A),A) ) ), file(uproots,k1_uproots), []). 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_k24_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k24_binop_2(A,B),k1_numbers,k5_numbers) ) ), file(binop_2,k24_binop_2), []). 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_funcop_1,axiom,( $true ), file(funcop_1,k2_funcop_1), []). fof(dt_k2_newton,axiom,( $true ), file(newton,k2_newton), []). 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_newton,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & v4_ordinal2(B) ) => m1_subset_1(k3_newton(A,B),k1_numbers) ) ), file(newton,k3_newton), []). 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_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_tarski,axiom,( $true ), file(tarski,k4_tarski), []). fof(dt_k5_facirc_1,axiom,( ! [A] : m1_circcomb(k5_facirc_1(A),1) ), file(facirc_1,k5_facirc_1), []). fof(dt_k5_finseq_1,axiom,( $true ), file(finseq_1,k5_finseq_1), []). 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_relat_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_relat_1(B) ) => v1_relat_1(k5_relat_1(A,B)) ) ), file(relat_1,k5_relat_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_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k6_nat_1(A,B),k1_numbers,k5_numbers) ) ), file(nat_1,k6_nat_1), []). fof(dt_k7_nat_3,axiom,( ! [A,B] : ( ( v7_seqm_3(B) & v1_polynom1(B) & m1_pboole(B,A) ) => v4_ordinal2(k7_nat_3(A,B)) ) ), file(nat_3,k7_nat_3), []). fof(dt_k8_nat_3,axiom,( ! [A,B] : ( ( v7_seqm_3(B) & v1_polynom1(B) & m1_pboole(B,A) ) => m2_subset_1(k8_nat_3(A,B),k1_numbers,k5_numbers) ) ), file(nat_3,k8_nat_3), []). fof(dt_k8_polynom1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v7_seqm_3(A) ) => m2_subset_1(k8_polynom1(A,B),k1_numbers,k5_numbers) ) ), file(polynom1,k8_polynom1), []). 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_k9_polynom1,axiom,( ! [A,B,C] : ( ( v1_seq_1(B) & m1_pboole(B,A) & v1_seq_1(C) & m1_pboole(C,A) ) => m1_pboole(k9_polynom1(A,B,C),A) ) ), file(polynom1,k9_polynom1), []). fof(dt_m1_circcomb,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ! [B] : ( m1_circcomb(B,A) => ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) ) ) ), file(circcomb,m1_circcomb), []). 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_pboole,axiom,( ! [A,B] : ( m1_pboole(B,A) => ( v1_relat_1(B) & v1_funct_1(B) ) ) ), file(pboole,m1_pboole), []). fof(dt_m1_polynom1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(k14_polynom1(A))) ) => ! [C] : ( m1_polynom1(C,A,B) => ( v7_seqm_3(C) & v1_polynom1(C) & m1_pboole(C,A) ) ) ) ), file(polynom1,m1_polynom1), []). 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_m1_trees_4,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m1_trees_4(C,A,B) => m2_finseq_1(C,A) ) ) ), file(trees_4,m1_trees_4), []). 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_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(existence_m1_circcomb,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ? [B] : m1_circcomb(B,A) ) ), file(circcomb,m1_circcomb), []). 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_pboole,axiom,( ! [A] : ? [B] : m1_pboole(B,A) ), file(pboole,m1_pboole), []). fof(existence_m1_polynom1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(k14_polynom1(A))) ) => ? [C] : m1_polynom1(C,A,B) ) ), file(polynom1,m1_polynom1), []). 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_m1_trees_4,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ? [C] : m1_trees_4(C,A,B) ) ), file(trees_4,m1_trees_4), []). 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_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(fc10_finset_1,axiom,( ! [A,B] : ( v1_finset_1(B) => v1_finset_1(k3_xboole_0(A,B)) ) ), file(finset_1,fc10_finset_1), []). 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_relat_1,axiom,( ! [A,B] : ( ( v1_xboole_0(A) & v1_relat_1(B) ) => ( v1_xboole_0(k5_relat_1(B,A)) & v1_relat_1(k5_relat_1(B,A)) ) ) ), file(relat_1,fc10_relat_1), []). 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_finset_1,axiom,( ! [A,B] : ( v1_finset_1(A) => v1_finset_1(k3_xboole_0(A,B)) ) ), file(finset_1,fc11_finset_1), []). 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_polynom1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( v1_relat_1(k5_finseq_1(A)) & v1_funct_1(k5_finseq_1(A)) & ~ v1_xboole_0(k5_finseq_1(A)) & v1_finset_1(k5_finseq_1(A)) & v1_finseq_1(k5_finseq_1(A)) & v1_funcop_1(k5_finseq_1(A)) & v1_matrlin(k5_finseq_1(A)) ) ) ), file(polynom1,fc11_polynom1), []). 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_relat_1,axiom, ( v1_xboole_0(k1_xboole_0) & v1_relat_1(k1_xboole_0) & v3_relat_1(k1_xboole_0) ), file(relat_1,fc12_relat_1), []). 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_finset_1,axiom,( ! [A,B] : ( ( v1_finset_1(A) & v1_finset_1(B) ) => v1_finset_1(k2_zfmisc_1(A,B)) ) ), file(finset_1,fc14_finset_1), []). 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(fc14_polynom1,axiom, ( 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_xboole_0(k1_xboole_0) & v1_finset_1(k1_xboole_0) & v1_finseq_1(k1_xboole_0) & v1_funcop_1(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) & v1_seq_1(k1_xboole_0) & v1_matrlin(k1_xboole_0) & v7_seqm_3(k1_xboole_0) ), file(polynom1,fc14_polynom1), []). 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(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(fc17_nat_3,axiom,( ! [A] : ( v4_ordinal2(A) => ( v1_relat_1(k11_nat_3(A)) & v1_funct_1(k11_nat_3(A)) & v1_seq_1(k11_nat_3(A)) & v7_seqm_3(k11_nat_3(A)) & v3_facirc_1(k11_nat_3(A)) ) ) ), file(nat_3,fc17_nat_3), []). fof(fc17_polynom1,axiom,( ! [A,B,C] : ( ( v1_seq_1(B) & m1_pboole(B,A) & v1_seq_1(C) & m1_pboole(C,A) ) => ( v1_relat_1(k9_polynom1(A,B,C)) & v1_funct_1(k9_polynom1(A,B,C)) & v1_seq_1(k9_polynom1(A,B,C)) ) ) ), file(polynom1,fc17_polynom1), []). fof(fc18_nat_3,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v4_ordinal2(A) ) => ( v1_relat_1(k11_nat_3(A)) & v1_funct_1(k11_nat_3(A)) & v1_seq_1(k11_nat_3(A)) & v7_seqm_3(k11_nat_3(A)) & v1_polynom1(k11_nat_3(A)) & v3_facirc_1(k11_nat_3(A)) ) ) ), file(nat_3,fc18_nat_3), []). fof(fc18_polynom1,axiom,( ! [A,B,C] : ( ( v7_seqm_3(B) & m1_pboole(B,A) & v7_seqm_3(C) & m1_pboole(C,A) ) => ( v1_relat_1(k9_polynom1(A,B,C)) & v1_funct_1(k9_polynom1(A,B,C)) & v1_seq_1(k9_polynom1(A,B,C)) & v7_seqm_3(k9_polynom1(A,B,C)) ) ) ), file(polynom1,fc18_polynom1), []). fof(fc19_nat_3,axiom, ( v1_xboole_0(k11_polynom1(k11_nat_3(1))) & v1_relat_1(k11_polynom1(k11_nat_3(1))) & v1_funct_1(k11_polynom1(k11_nat_3(1))) & v2_funct_1(k11_polynom1(k11_nat_3(1))) & v1_finset_1(k11_polynom1(k11_nat_3(1))) & v1_membered(k11_polynom1(k11_nat_3(1))) & v2_membered(k11_polynom1(k11_nat_3(1))) & v3_membered(k11_polynom1(k11_nat_3(1))) & v4_membered(k11_polynom1(k11_nat_3(1))) & v5_membered(k11_polynom1(k11_nat_3(1))) & v1_polynom1(k11_polynom1(k11_nat_3(1))) ), file(nat_3,fc19_nat_3), []). fof(fc1_finset_1,axiom,( ! [A] : ( ~ v1_xboole_0(k1_tarski(A)) & v1_finset_1(k1_tarski(A)) ) ), file(finset_1,fc1_finset_1), []). fof(fc1_funct_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_relat_1(B) & v1_funct_1(B) ) => ( v1_relat_1(k5_relat_1(A,B)) & v1_funct_1(k5_relat_1(A,B)) ) ) ), file(funct_1,fc1_funct_1), []). fof(fc1_int_1,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v1_int_1(k2_xcmplx_0(A,B)) ) ) ), file(int_1,fc1_int_1), []). fof(fc1_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v4_ordinal2(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(nat_1,fc1_nat_1), []). fof(fc1_newton,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v4_ordinal2(B) ) => ( v1_xcmplx_0(k2_newton(A,B)) & v1_xreal_0(k2_newton(A,B)) ) ) ), file(newton,fc1_newton), []). fof(fc1_pre_circ,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_finset_1(A) & v5_membered(A) ) => ( v4_ordinal2(k2_seq_4(A)) & v1_xcmplx_0(k2_seq_4(A)) & v1_xreal_0(k2_seq_4(A)) ) ) ), file(pre_circ,fc1_pre_circ), []). fof(fc1_relat_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_relat_1(B) ) => v1_relat_1(k3_xboole_0(A,B)) ) ), file(relat_1,fc1_relat_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(fc1_tex_2,axiom,( ! [A] : ( ~ v1_xboole_0(k1_tarski(A)) & v1_realset1(k1_tarski(A)) & v1_finset_1(k1_tarski(A)) ) ), file(tex_2,fc1_tex_2), []). fof(fc20_nat_3,axiom,( ! [A,B] : ( ( v1_int_2(A) & m1_subset_1(A,k5_numbers) & ~ v1_xboole_0(B) & v4_ordinal2(B) ) => ( ~ v1_xboole_0(k11_polynom1(k11_nat_3(k3_newton(A,B)))) & v1_finset_1(k11_polynom1(k11_nat_3(k3_newton(A,B)))) & v1_realset1(k11_polynom1(k11_nat_3(k3_newton(A,B)))) ) ) ), file(nat_3,fc20_nat_3), []). fof(fc20_polynom1,axiom, ( 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_xboole_0(k1_xboole_0) & v1_finset_1(k1_xboole_0) & v1_finseq_1(k1_xboole_0) & v1_funcop_1(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) & v1_seq_1(k1_xboole_0) & v1_matrlin(k1_xboole_0) & v7_seqm_3(k1_xboole_0) & v1_polynom1(k1_xboole_0) ), file(polynom1,fc20_polynom1), []). fof(fc21_nat_3,axiom,( ! [A] : ( ( v1_int_2(A) & m1_subset_1(A,k5_numbers) ) => ( ~ v1_xboole_0(k11_polynom1(k11_nat_3(A))) & v1_finset_1(k11_polynom1(k11_nat_3(A))) & v1_realset1(k11_polynom1(k11_nat_3(A))) ) ) ), file(nat_3,fc21_nat_3), []). fof(fc21_polynom1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_polynom1(A) ) => v1_finset_1(k11_polynom1(A)) ) ), file(polynom1,fc21_polynom1), []). 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_nat_3,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v4_ordinal2(A) ) => ( v1_relat_1(k12_nat_3(A)) & v1_funct_1(k12_nat_3(A)) & v1_seq_1(k12_nat_3(A)) & v7_seqm_3(k12_nat_3(A)) & v1_polynom1(k12_nat_3(A)) & v3_facirc_1(k12_nat_3(A)) ) ) ), file(nat_3,fc22_nat_3), []). 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_polynom1,axiom,( ! [A,B,C] : ( ( v7_seqm_3(B) & v1_polynom1(B) & m1_pboole(B,A) & v7_seqm_3(C) & v1_polynom1(C) & m1_pboole(C,A) ) => ( v1_relat_1(k9_polynom1(A,B,C)) & v1_funct_1(k9_polynom1(A,B,C)) & v1_seq_1(k9_polynom1(A,B,C)) & v7_seqm_3(k9_polynom1(A,B,C)) & v1_polynom1(k9_polynom1(A,B,C)) ) ) ), file(polynom1,fc23_polynom1), []). 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(fc25_polynom1,axiom,( ! [A] : ~ v1_xboole_0(k13_polynom1(A)) ), file(polynom1,fc25_polynom1), []). fof(fc27_membered,axiom,( ! [A,B] : ( v1_membered(A) => v1_membered(k3_xboole_0(A,B)) ) ), file(membered,fc27_membered), []). fof(fc28_membered,axiom,( ! [A,B] : ( v1_membered(A) => v1_membered(k3_xboole_0(B,A)) ) ), file(membered,fc28_membered), []). 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(fc2_finset_1,axiom,( ! [A,B] : ( ~ v1_xboole_0(k2_tarski(A,B)) & v1_finset_1(k2_tarski(A,B)) ) ), file(finset_1,fc2_finset_1), []). fof(fc2_int_1,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & v1_int_1(k3_xcmplx_0(A,B)) ) ) ), file(int_1,fc2_int_1), []). fof(fc2_membered,axiom, ( ~ v1_xboole_0(k1_numbers) & v1_membered(k1_numbers) & v2_membered(k1_numbers) ), file(membered,fc2_membered), []). fof(fc2_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v4_ordinal2(k3_xcmplx_0(A,B)) & v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(nat_1,fc2_nat_1), []). fof(fc2_nat_3,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v4_ordinal2(A) & v4_ordinal2(B) ) => ( ~ v1_xboole_0(k2_newton(A,B)) & v4_ordinal2(k2_newton(A,B)) & v1_xreal_0(k2_newton(A,B)) & v2_xreal_0(k2_newton(A,B)) & ~ v3_xreal_0(k2_newton(A,B)) & v1_int_1(k2_newton(A,B)) & v1_xcmplx_0(k2_newton(A,B)) ) ) ), file(nat_3,fc2_nat_3), []). fof(fc2_newton,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v4_ordinal2(k2_newton(A,B)) & v1_xcmplx_0(k2_newton(A,B)) & v1_xreal_0(k2_newton(A,B)) & ~ v3_xreal_0(k2_newton(A,B)) & v1_int_1(k2_newton(A,B)) ) ) ), file(newton,fc2_newton), []). fof(fc2_xcmplx_0,axiom,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => v1_xcmplx_0(k2_xcmplx_0(A,B)) ) ), file(xcmplx_0,fc2_xcmplx_0), []). 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(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_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & ~ v1_xboole_0(B) & v4_ordinal2(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v4_ordinal2(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(nat_1,fc3_nat_1), []). fof(fc3_polynom1,axiom, ( 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_xboole_0(k1_xboole_0) & v1_finset_1(k1_xboole_0) & v1_finseq_1(k1_xboole_0) & v1_funcop_1(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) & v1_matrlin(k1_xboole_0) ), file(polynom1,fc3_polynom1), []). fof(fc3_xcmplx_0,axiom,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => v1_xcmplx_0(k3_xcmplx_0(A,B)) ) ), file(xcmplx_0,fc3_xcmplx_0), []). 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_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & ~ v1_xboole_0(B) & v4_ordinal2(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v4_ordinal2(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(nat_1,fc4_nat_1), []). fof(fc4_relat_1,axiom, ( v1_xboole_0(k1_xboole_0) & v1_relat_1(k1_xboole_0) ), file(relat_1,fc4_relat_1), []). 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_newton,axiom, ( ~ v1_xboole_0(k12_newton) & ~ v1_finset_1(k12_newton) & v1_membered(k12_newton) & v2_membered(k12_newton) & v3_membered(k12_newton) & v4_membered(k12_newton) & v5_membered(k12_newton) ), file(newton,fc5_newton), []). fof(fc5_polynom1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & v1_matrlin(A) ) => ( v1_relat_1(k1_funct_1(A,B)) & v1_funct_1(k1_funct_1(A,B)) & v1_finset_1(k1_funct_1(A,B)) & v1_finseq_1(k1_funct_1(A,B)) ) ) ), file(polynom1,fc5_polynom1), []). fof(fc5_relat_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_relat_1(A) ) => ~ v1_xboole_0(k1_relat_1(A)) ) ), file(relat_1,fc5_relat_1), []). fof(fc6_int_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_int_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v1_int_1(k2_xcmplx_0(B,A)) ) ) ), file(int_1,fc6_int_1), []). 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(fc7_circcmb3,axiom, ( ~ v1_xboole_0(k5_ordinal2) & v1_ordinal1(k5_ordinal2) & v2_ordinal1(k5_ordinal2) & v3_ordinal1(k5_ordinal2) & v1_membered(k5_ordinal2) & v2_membered(k5_ordinal2) & v3_membered(k5_ordinal2) & v4_membered(k5_ordinal2) & v5_membered(k5_ordinal2) & ~ v2_facirc_1(k5_ordinal2) ), file(circcmb3,fc7_circcmb3), []). fof(fc7_int_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_int_1(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(B,A)) & v1_xreal_0(k3_xcmplx_0(B,A)) & v1_int_1(k3_xcmplx_0(B,A)) ) ) ), file(int_1,fc7_int_1), []). fof(fc7_membered,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_membered(k1_tarski(A)) ) ), file(membered,fc7_membered), []). fof(fc7_relat_1,axiom,( ! [A] : ( v1_xboole_0(A) => ( v1_xboole_0(k1_relat_1(A)) & v1_relat_1(k1_relat_1(A)) ) ) ), file(relat_1,fc7_relat_1), []). 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_xcmplx_0,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & ~ v1_xboole_0(B) & v1_xcmplx_0(B) ) => ( ~ v1_xboole_0(k3_xcmplx_0(A,B)) & v1_xcmplx_0(k3_xcmplx_0(A,B)) ) ) ), file(xcmplx_0,fc8_xcmplx_0), []). 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_relat_1,axiom,( ! [A,B] : ( ( v1_xboole_0(A) & v1_relat_1(B) ) => ( v1_xboole_0(k5_relat_1(A,B)) & v1_relat_1(k5_relat_1(A,B)) ) ) ), file(relat_1,fc9_relat_1), []). 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(fraenkel_a_1_0_finseq_1,axiom,( ! [A,B] : ( v4_ordinal2(B) => ( r2_hidden(A,a_1_0_finseq_1(B)) <=> ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & A = C & r1_xreal_0(1,C) & r1_xreal_0(C,B) ) ) ) ), file(finseq_1,a_1_0_finseq_1), []). 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(idempotence_k6_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k6_nat_1(A,A) = A ) ), file(nat_1,k6_nat_1), []). fof(rc10_polynom1,axiom,( ! [A] : ? [B] : ( m1_pboole(B,A) & v1_relat_1(B) & v1_funct_1(B) & v1_seq_1(B) ) ), file(polynom1,rc10_polynom1), []). fof(rc11_polynom1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & ~ v1_xboole_0(A) & v1_seq_1(A) & v7_seqm_3(A) & v1_polynom1(A) ) ), file(polynom1,rc11_polynom1), []). fof(rc13_polynom1,axiom,( ! [A] : ? [B] : ( m1_pboole(B,A) & v1_relat_1(B) & v1_funct_1(B) & v1_seq_1(B) & v7_seqm_3(B) & v1_polynom1(B) ) ), file(polynom1,rc13_polynom1), []). fof(rc1_card_5,axiom,( ? [A] : ~ v1_finset_1(A) ), file(card_5,rc1_card_5), []). fof(rc1_finset_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) ) ), file(finset_1,rc1_finset_1), []). fof(rc1_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc1_funct_1), []). fof(rc1_int_1,axiom,( ? [A] : ( m1_subset_1(A,k1_numbers) & v1_xcmplx_0(A) & v1_xreal_0(A) & v1_int_1(A) ) ), file(int_1,rc1_int_1), []). 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_nat_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc1_nat_1), []). fof(rc1_nat_3,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & v1_seq_1(A) & v7_seqm_3(A) & v3_facirc_1(A) ) ), file(nat_3,rc1_nat_3), []). fof(rc1_newton,axiom,( ? [A] : ( m1_subset_1(A,k5_numbers) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_int_1(A) & v1_int_2(A) ) ), file(newton,rc1_newton), []). fof(rc1_pre_circ,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) & v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ), file(pre_circ,rc1_pre_circ), []). fof(rc1_relat_1,axiom,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) ) ), file(relat_1,rc1_relat_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_tex_2,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_realset1(A) ) ), file(tex_2,rc1_tex_2), []). fof(rc1_xcmplx_0,axiom,( ? [A] : v1_xcmplx_0(A) ), file(xcmplx_0,rc1_xcmplx_0), []). fof(rc1_xreal_0,axiom,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(xreal_0,rc1_xreal_0), []). fof(rc2_finset_1,axiom,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v2_funct_1(B) & v1_ordinal1(B) & v2_ordinal1(B) & v3_ordinal1(B) & v4_ordinal2(B) & v1_finset_1(B) ) ), file(finset_1,rc2_finset_1), []). fof(rc2_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) ), file(funct_1,rc2_funct_1), []). fof(rc2_int_1,axiom,( ? [A] : v1_int_1(A) ), file(int_1,rc2_int_1), []). fof(rc2_nat_1,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(A) & v3_ordinal1(A) ) ), file(nat_1,rc2_nat_1), []). fof(rc2_nat_3,axiom,( ! [A] : ? [B] : ( m1_pboole(B,A) & v1_relat_1(B) & v1_funct_1(B) & v1_seq_1(B) & v7_seqm_3(B) & v1_polynom1(B) & v3_facirc_1(B) ) ), file(nat_3,rc2_nat_3), []). fof(rc2_polynom1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) & v1_funcop_1(A) & v1_matrlin(A) ) ), file(polynom1,rc2_polynom1), []). fof(rc2_relat_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) ) ), file(relat_1,rc2_relat_1), []). fof(rc2_tex_2,axiom,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_tex_2(B,k1_zfmisc_1(A)) ) ), file(tex_2,rc2_tex_2), []). fof(rc2_xcmplx_0,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) ) ), file(xcmplx_0,rc2_xcmplx_0), []). 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_finset_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & v1_finset_1(B) ) ) ), file(finset_1,rc3_finset_1), []). fof(rc3_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ), file(funct_1,rc3_funct_1), []). fof(rc3_nat_1,axiom,( ? [A] : ( m1_subset_1(A,k5_numbers) & ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc3_nat_1), []). fof(rc3_polynom1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) & v1_card_3(A) ) ), file(polynom1,rc3_polynom1), []). fof(rc3_relat_1,axiom,( ? [A] : ( v1_relat_1(A) & v3_relat_1(A) ) ), file(relat_1,rc3_relat_1), []). fof(rc3_tex_2,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_tex_2(B,k1_zfmisc_1(A)) ) ) ), file(tex_2,rc3_tex_2), []). 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_finset_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & v1_finset_1(B) ) ) ), file(finset_1,rc4_finset_1), []). fof(rc4_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v3_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc4_funct_1), []). fof(rc4_polynom1,axiom,( ? [A] : ( m1_finseq_1(A,k5_numbers) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) & v1_card_3(A) ) ), file(polynom1,rc4_polynom1), []). fof(rc4_tex_2,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & ~ v1_tex_2(B,k1_zfmisc_1(A)) ) ) ), file(tex_2,rc4_tex_2), []). 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(rc5_tex_2,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & v1_realset1(B) ) ) ), file(tex_2,rc5_tex_2), []). fof(rc6_circcmb3,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) & v3_facirc_1(A) & v7_seqm_3(A) ) ), file(circcmb3,rc6_circcmb3), []). fof(rc6_tex_2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & ~ v1_realset1(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & v1_realset1(B) & v1_tex_2(B,k1_zfmisc_1(A)) ) ) ), file(tex_2,rc6_tex_2), []). fof(rc7_circcmb3,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ? [B] : ( m1_circcomb(B,A) & v1_relat_1(B) & v1_funct_1(B) & v2_funct_1(B) & v1_finset_1(B) & v1_finseq_1(B) & v3_facirc_1(B) & v7_seqm_3(B) ) ) ), file(circcmb3,rc7_circcmb3), []). fof(rc7_polynom1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) & v7_seqm_3(A) ) ), file(polynom1,rc7_polynom1), []). fof(rc7_tex_2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & ~ v1_realset1(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & ~ v1_realset1(B) & ~ v1_tex_2(B,k1_zfmisc_1(A)) ) ) ), file(tex_2,rc7_tex_2), []). fof(rc8_polynom1,axiom,( ! [A] : ? [B] : ( m1_pboole(B,A) & v1_relat_1(B) & v1_funct_1(B) & v1_seq_1(B) & v7_seqm_3(B) ) ), file(polynom1,rc8_polynom1), []). fof(rc9_polynom1,axiom,( ! [A] : ? [B] : ( m1_pboole(B,A) & v1_relat_1(B) & v1_funct_1(B) & v1_seq_1(B) ) ), file(polynom1,rc9_polynom1), []). fof(redefinition_k10_wsierp_1,axiom,( ! [A] : ( m1_finseq_1(A,k5_numbers) => k10_wsierp_1(A) = k16_rvsum_1(A) ) ), file(wsierp_1,k10_wsierp_1), []). 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_k13_binarith,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => k13_binarith(A,B) = k5_finseq_1(B) ) ), file(binarith,k13_binarith), []). fof(redefinition_k14_polynom1,axiom,( ! [A] : k14_polynom1(A) = k13_polynom1(A) ), file(polynom1,k14_polynom1), []). fof(redefinition_k1_polynom2,axiom,( ! [A,B] : ( ( v7_seqm_3(B) & v1_polynom1(B) & m1_pboole(B,A) ) => k1_polynom2(A,B) = k11_polynom1(B) ) ), file(polynom2,k1_polynom2), []). fof(redefinition_k1_pre_circ,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_finset_1(A) & v2_membered(A) ) => k1_pre_circ(A) = k2_seq_4(A) ) ), file(pre_circ,k1_pre_circ), []). 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_k24_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k24_binop_2(A,B) = k3_xcmplx_0(A,B) ) ), file(binop_2,k24_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_k3_newton,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & v4_ordinal2(B) ) => k3_newton(A,B) = k2_newton(A,B) ) ), file(newton,k3_newton), []). fof(redefinition_k5_facirc_1,axiom,( ! [A] : k5_facirc_1(A) = k5_finseq_1(A) ), file(facirc_1,k5_facirc_1), []). fof(redefinition_k5_numbers,axiom,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), []). 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_k8_nat_3,axiom,( ! [A,B] : ( ( v7_seqm_3(B) & v1_polynom1(B) & m1_pboole(B,A) ) => k8_nat_3(A,B) = k7_nat_3(A,B) ) ), file(nat_3,k8_nat_3), []). fof(redefinition_k8_polynom1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v7_seqm_3(A) ) => k8_polynom1(A,B) = k1_funct_1(A,B) ) ), file(polynom1,k8_polynom1), []). fof(redefinition_k9_finseq_1,axiom,( ! [A] : k9_finseq_1(A) = k5_finseq_1(A) ), file(finseq_1,k9_finseq_1), []). fof(redefinition_m1_polynom1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(k14_polynom1(A))) ) => ! [C] : ( m1_polynom1(C,A,B) <=> m1_subset_1(C,B) ) ) ), file(polynom1,m1_polynom1), []). fof(redefinition_m1_trees_4,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m1_trees_4(C,A,B) <=> m1_finseq_1(C,B) ) ) ), file(trees_4,m1_trees_4), []). 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_r6_pboole,axiom,( ! [A,B,C] : ( ( m1_pboole(B,A) & m1_pboole(C,A) ) => ( r6_pboole(A,B,C) <=> B = C ) ) ), file(pboole,r6_pboole), []). fof(reflexivity_r1_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => r1_nat_1(A,A) ) ), file(nat_1,r1_nat_1), []). 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(reflexivity_r6_pboole,axiom,( ! [A,B,C] : ( ( m1_pboole(B,A) & m1_pboole(C,A) ) => r6_pboole(A,B,B) ) ), file(pboole,r6_pboole), []). fof(s2_nat_1__e3_101__nat_3,axiom, ( ( ! [A] : ( ( ~ v1_xboole_0(A) & v4_ordinal2(A) ) => ( r1_tarski(k1_polynom2(k12_newton,k12_nat_3(A)),k2_finseq_1(0)) => k8_nat_3(k12_newton,k12_nat_3(A)) = A ) ) & ! [B] : ( v4_ordinal2(B) => ( ! [C] : ( ( ~ v1_xboole_0(C) & v4_ordinal2(C) ) => ( r1_tarski(k1_polynom2(k12_newton,k12_nat_3(C)),k2_finseq_1(B)) => k8_nat_3(k12_newton,k12_nat_3(C)) = C ) ) => ! [D] : ( ( ~ v1_xboole_0(D) & v4_ordinal2(D) ) => ( r1_tarski(k1_polynom2(k12_newton,k12_nat_3(D)),k2_finseq_1(k2_xcmplx_0(B,1))) => k8_nat_3(k12_newton,k12_nat_3(D)) = D ) ) ) ) ) => ! [B] : ( v4_ordinal2(B) => ! [E] : ( ( ~ v1_xboole_0(E) & v4_ordinal2(E) ) => ( r1_tarski(k1_polynom2(k12_newton,k12_nat_3(E)),k2_finseq_1(B)) => k8_nat_3(k12_newton,k12_nat_3(E)) = E ) ) ) ), file(nat_3,s2_nat_1__e3_101__nat_3), []). 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(symmetry_r1_xboole_0,axiom,( ! [A,B] : ( r1_xboole_0(A,B) => r1_xboole_0(B,A) ) ), file(xboole_0,r1_xboole_0), []). fof(symmetry_r2_int_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => ( r2_int_2(A,B) => r2_int_2(B,A) ) ) ), file(int_2,r2_int_2), []). fof(symmetry_r6_pboole,axiom,( ! [A,B,C] : ( ( m1_pboole(B,A) & m1_pboole(C,A) ) => ( r6_pboole(A,B,C) => r6_pboole(A,C,B) ) ) ), file(pboole,r6_pboole), []). fof(t11_newton,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v1_xreal_0(B) => k2_newton(B,k2_xcmplx_0(A,1)) = k3_xcmplx_0(k2_newton(B,A),B) ) ) ), file(newton,t11_newton), []). fof(t125_rvsum_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => k16_rvsum_1(k12_finseq_1(k1_numbers,A)) = A ) ), file(rvsum_1,t125_rvsum_1), []). fof(t19_nat_3,axiom,( ! [A,B] : ( ( v7_seqm_3(B) & v1_polynom1(B) & m1_pboole(B,A) ) => ! [C] : ( ( v7_seqm_3(C) & v1_polynom1(C) & m1_pboole(C,A) ) => ( r1_xboole_0(k1_polynom2(A,B),k1_polynom2(A,C)) => k8_nat_3(A,k9_polynom1(A,B,C)) = k24_binop_2(k8_nat_3(A,B),k8_nat_3(A,C)) ) ) ) ), file(nat_3,t19_nat_3), []). fof(t1_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k2_xcmplx_0(A,0) = A ) ), file(arithm,t1_arithm), []). 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_uproots,axiom,( ! [A] : ( v4_ordinal2(A) => ( ~ ( ~ v1_xboole_0(A) & A != 1 & r1_xreal_0(A,1) ) & ~ ( ~ ( A != 1 & r1_xreal_0(A,1) ) & v1_xboole_0(A) ) ) ) ), file(uproots,t1_uproots), []). 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(t20_bagorder,axiom,( ! [A,B] : ( ( v7_seqm_3(B) & v1_polynom1(B) & m1_pboole(B,A) ) => ( k1_polynom2(A,B) = k1_xboole_0 => r6_pboole(A,B,k16_polynom1(A)) ) ) ), file(bagorder,t20_bagorder), []). fof(t20_nat_3,axiom,( ! [A] : k8_nat_3(A,k16_polynom1(A)) = 1 ), file(nat_3,t20_nat_3), []). 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(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(t2_tarski,axiom,( ! [A,B] : ( ! [C] : ( r2_hidden(C,A) <=> r2_hidden(C,B) ) => A = B ) ), file(tarski,t2_tarski), []). fof(t2_xboole_1,axiom,( ! [A] : r1_tarski(k1_xboole_0,A) ), file(xboole_1,t2_xboole_1), []). fof(t34_nat_3,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( r2_hidden(B,k11_polynom1(k11_nat_3(A))) => ( v1_int_2(B) & m2_subset_1(B,k1_numbers,k5_numbers) ) ) ) ), file(nat_3,t34_nat_3), []). fof(t36_circcmb3,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( r2_hidden(B,k1_relat_1(A)) => k5_relat_1(k5_facirc_1(B),A) = k5_facirc_1(k1_funct_1(A,B)) ) ) ), file(circcmb3,t36_circcmb3), []). fof(t37_nat_3,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( ( v1_int_2(B) & m2_subset_1(B,k1_numbers,k5_numbers) & r1_nat_1(B,A) ) => ( v1_xboole_0(A) | r2_hidden(B,k11_polynom1(k11_nat_3(A))) ) ) ) ) ), file(nat_3,t37_nat_3), []). 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(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_subset,axiom,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), []). fof(t42_nat_3,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( ( v1_int_2(B) & m2_subset_1(B,k1_numbers,k5_numbers) ) => ( A != 0 => k1_polynom2(k12_newton,k11_nat_3(k3_newton(B,A))) = k1_tarski(B) ) ) ) ), file(nat_3,t42_nat_3), []). fof(t45_nat_3,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v4_ordinal2(A) ) => ! [B] : ( ( ~ v1_xboole_0(B) & v4_ordinal2(B) ) => r1_tarski(k1_polynom2(k12_newton,k11_nat_3(A)),k1_polynom2(k12_newton,k11_nat_3(k3_xcmplx_0(A,B)))) ) ) ), file(nat_3,t45_nat_3), []). fof(t48_int_2,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(2,A) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ ( v1_int_2(B) & r1_nat_1(B,A) ) ) ) ) ), file(int_2,t48_int_2), []). 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_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(t51_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ! [C] : ( v4_ordinal2(C) => ( ( r1_nat_1(A,B) & r1_nat_1(B,C) ) => r1_nat_1(A,C) ) ) ) ) ), file(nat_1,t51_nat_1), []). fof(t58_nat_3,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & m2_subset_1(A,k1_numbers,k5_numbers) ) => ! [B] : ( ( ~ v1_xboole_0(B) & m2_subset_1(B,k1_numbers,k5_numbers) ) => ( r2_int_2(A,B) => r6_pboole(k12_newton,k12_nat_3(k24_binop_2(A,B)),k9_polynom1(k12_newton,k12_nat_3(A),k12_nat_3(B))) ) ) ) ), file(nat_3,t58_nat_3), []). fof(t59_nat_3,axiom,( ! [A] : ( ( v1_int_2(A) & m2_subset_1(A,k1_numbers,k5_numbers) ) => ! [B] : ( ( ~ v1_xboole_0(B) & v4_ordinal2(B) ) => k8_polynom1(k12_nat_3(k3_newton(A,B)),A) = k3_newton(A,B) ) ) ), file(nat_3,t59_nat_3), []). fof(t5_int_2,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( ( A = 0 & B = 0 ) <=> k6_nat_1(A,B) = 0 ) ) ) ), file(int_2,t5_int_2), []). fof(t5_nat_3,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ! [C] : ( ( v1_int_2(C) & m2_subset_1(C,k1_numbers,k5_numbers) ) => ( r1_nat_1(C,k2_newton(A,B)) => r1_nat_1(C,A) ) ) ) ) ), file(nat_3,t5_nat_3), []). 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(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(t6_uproots,axiom,( ! [A] : k1_uproots(k1_tarski(A)) = k9_finseq_1(A) ), file(uproots,t6_uproots), []). fof(t6_xcmplx_1,axiom,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => ~ ( k3_xcmplx_0(A,B) = 0 & A != 0 & B != 0 ) ) ) ), file(xcmplx_1,t6_xcmplx_1), []). fof(t7_boole,axiom,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), []). 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(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), []).