% Mizar problem: t65_chain_1,chain_1,3003,18 fof(t65_chain_1,conjecture,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( ~ v1_xboole_0(B) & m2_subset_1(B,k1_numbers,k5_numbers) ) => ! [C] : ( m1_chain_1(C,B) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(k5_chain_1(B,C,k1_nat_1(k1_nat_1(A,1),1)))) => k11_chain_1(B,C,A,k11_chain_1(B,C,k1_nat_1(A,1),D)) = k6_chain_1(B,C,A) ) ) ) ) ), 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_abian,cc1_afinsq_1,cc1_finseq_1,cc1_finseq_5,cc1_finset_1,cc1_membered,cc1_nat_1,cc1_realset1,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_nat_1,cc3_membered,cc3_nat_1,cc4_membered,cc6_membered,cc9_membered,commutativity_k1_nat_1,commutativity_k2_nat_1,commutativity_k2_tarski,commutativity_k2_xboole_0,commutativity_k2_xcmplx_0,commutativity_k3_xboole_0,commutativity_k3_xcmplx_0,commutativity_k4_subset_1,commutativity_k5_subset_1,commutativity_k5_xboole_0,commutativity_k8_chain_1,connectedness_r1_xreal_0,d12_chain_1,d13_chain_1,d16_chain_1,d1_tarski,d1_xboole_0,d2_xboole_0,d3_xboole_0,d4_chain_1,d4_xboole_0,d5_tarski,d7_chain_1,d9_chain_1,dt_k10_chain_1,dt_k11_chain_1,dt_k1_card_1,dt_k1_domain_1,dt_k1_euclid,dt_k1_finseq_1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_chain_1,dt_k2_finseq_1,dt_k2_nat_1,dt_k2_tarski,dt_k2_xboole_0,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k3_chain_1,dt_k3_subset_1,dt_k3_xboole_0,dt_k3_xcmplx_0,dt_k4_card_1,dt_k4_chain_1,dt_k4_subset_1,dt_k4_tarski,dt_k4_xboole_0,dt_k5_chain_1,dt_k5_numbers,dt_k5_ordinal2,dt_k5_real_1,dt_k5_subset_1,dt_k5_xboole_0,dt_k6_chain_1,dt_k6_domain_1,dt_k6_subset_1,dt_k6_xcmplx_0,dt_k8_chain_1,dt_k8_funct_2,dt_k9_chain_1,dt_m1_chain_1,dt_m1_finseq_1,dt_m1_finseq_2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_chain_1,dt_m2_finseq_1,dt_m2_finseq_2,dt_m2_relset_1,dt_m2_subset_1,dt_m3_chain_1,existence_m1_chain_1,existence_m1_finseq_1,existence_m1_finseq_2,existence_m1_relset_1,existence_m1_subset_1,existence_m2_chain_1,existence_m2_finseq_1,existence_m2_finseq_2,existence_m2_relset_1,existence_m2_subset_1,existence_m3_chain_1,fc10_abian,fc10_finset_1,fc10_membered,fc11_abian,fc11_finset_1,fc11_membered,fc12_abian,fc12_finset_1,fc12_membered,fc13_abian,fc13_membered,fc14_abian,fc14_finset_1,fc14_membered,fc15_abian,fc15_membered,fc16_membered,fc17_abian,fc17_finset_1,fc1_abian,fc1_chain_1,fc1_finseq_1,fc1_finset_1,fc1_nat_1,fc1_subset_1,fc22_membered,fc23_membered,fc24_membered,fc25_membered,fc26_membered,fc27_membered,fc28_membered,fc29_membered,fc2_abian,fc2_afinsq_1,fc2_chain_1,fc2_finseq_1,fc2_finset_1,fc2_membered,fc2_nat_1,fc2_subset_1,fc30_membered,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc37_membered,fc38_membered,fc39_membered,fc3_abian,fc3_chain_1,fc3_nat_1,fc3_realset1,fc3_subset_1,fc40_membered,fc41_membered,fc42_membered,fc43_membered,fc44_membered,fc45_membered,fc46_membered,fc4_abian,fc4_chain_1,fc4_nat_1,fc4_subset_1,fc5_abian,fc5_membered,fc6_abian,fc6_membered,fc7_abian,fc7_membered,fc8_abian,fc8_membered,fc9_abian,fc9_finset_1,fc9_membered,fraenkel_a_3_0_chain_1,fraenkel_a_4_0_chain_1,fraenkel_a_4_2_chain_1,idempotence_k2_xboole_0,idempotence_k3_xboole_0,idempotence_k4_subset_1,idempotence_k5_subset_1,involutiveness_k3_subset_1,rc1_abian,rc1_afinsq_1,rc1_chain_1,rc1_finseq_1,rc1_finseq_5,rc1_finset_1,rc1_membered,rc1_nat_1,rc1_realset1,rc1_subset_1,rc2_abian,rc2_chain_1,rc2_finseq_1,rc2_finseq_5,rc2_finset_1,rc2_nat_1,rc2_realset1,rc2_subset_1,rc3_abian,rc3_afinsq_1,rc3_chain_1,rc3_finseq_1,rc3_finset_1,rc3_nat_1,rc4_abian,rc4_afinsq_1,rc4_finseq_1,rc4_finset_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,redefinition_k1_domain_1,redefinition_k1_nat_1,redefinition_k2_chain_1,redefinition_k2_finseq_1,redefinition_k2_nat_1,redefinition_k3_chain_1,redefinition_k4_card_1,redefinition_k4_subset_1,redefinition_k5_numbers,redefinition_k5_real_1,redefinition_k5_subset_1,redefinition_k6_domain_1,redefinition_k6_subset_1,redefinition_k8_chain_1,redefinition_k8_funct_2,redefinition_m2_finseq_1,redefinition_m2_finseq_2,redefinition_m2_relset_1,redefinition_m2_subset_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,s3_funct_2__e11_92_1_1_1_1_2__chain_1,s3_funct_2__e14_92_1_1_1_1_2__chain_1,s3_funct_2__e26_92_1_1_1_1_2__chain_1,s3_funct_2__e29_92_1_1_1_1_2__chain_1,s4_chain_1__e10_92_1_1__chain_1,s4_chain_1__e5_92_1_1_4_1_2__chain_1,spc0_boole,spc1_boole,spc1_numerals,spc2_boole,spc2_numerals,symmetry_r1_xboole_0,t113_funct_2,t12_xboole_1,t1_arithm,t1_boole,t1_numerals,t1_real,t1_subset,t1_xboole_1,t29_chain_1,t2_arithm,t2_boole,t2_real,t2_subset,t2_tarski,t32_chain_1,t33_chain_1,t34_chain_1,t37_zfmisc_1,t38_nat_1,t39_xboole_1,t3_arithm,t3_boole,t3_real,t3_subset,t48_chain_1,t4_arithm,t4_boole,t4_real,t4_subset,t50_chain_1,t51_chain_1,t52_chain_1,t53_card_2,t53_chain_1,t54_chain_1,t54_subset_1,t56_zfmisc_1,t5_boole,t5_real,t5_subset,t60_card_2,t61_chain_1,t63_card_2,t63_chain_1,t64_chain_1,t6_boole,t6_chain_1,t6_real,t78_card_1,t79_card_1,t7_boole,t7_real,t7_xboole_1,t8_boole,t8_real,t8_xboole_1]), [file(chain_1,t65_chain_1)]). 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_abian,axiom,( ! [A] : ( v2_setfam_1(A) => v1_realset1(A) ) ), file(abian,cc1_abian), []). fof(cc1_afinsq_1,axiom,( ! [A] : ( v4_ordinal2(A) => v1_finset_1(A) ) ), file(afinsq_1,cc1_afinsq_1), []). fof(cc1_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) ) ), file(finseq_1,cc1_finseq_1), []). fof(cc1_finseq_5,axiom,( ! [A] : ( v1_xboole_0(A) => v1_realset1(A) ) ), file(finseq_5,cc1_finseq_5), []). fof(cc1_finset_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_finset_1(A) ) ), file(finset_1,cc1_finset_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_realset1,axiom,( ! [A] : ( ~ v1_realset1(A) => ~ v1_xboole_0(A) ) ), file(realset1,cc1_realset1), []). 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(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_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(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(cc4_membered,axiom,( ! [A] : ( v2_membered(A) => v1_membered(A) ) ), file(membered,cc4_membered), []). 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(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_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_k2_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k2_nat_1(A,B) = k2_nat_1(B,A) ) ), file(nat_1,k2_nat_1), []). fof(commutativity_k2_tarski,axiom,( ! [A,B] : k2_tarski(A,B) = k2_tarski(B,A) ), file(tarski,k2_tarski), []). fof(commutativity_k2_xboole_0,axiom,( ! [A,B] : k2_xboole_0(A,B) = k2_xboole_0(B,A) ), file(xboole_0,k2_xboole_0), []). 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_k4_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k4_subset_1(A,B,C) = k4_subset_1(A,C,B) ) ), file(subset_1,k4_subset_1), []). 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_k5_xboole_0,axiom,( ! [A,B] : k5_xboole_0(A,B) = k5_xboole_0(B,A) ), file(xboole_0,k5_xboole_0), []). fof(commutativity_k8_chain_1,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k5_numbers) & m1_chain_1(B,A) & m1_subset_1(C,k5_numbers) & m1_subset_1(D,k1_zfmisc_1(k5_chain_1(A,B,C))) & m1_subset_1(E,k1_zfmisc_1(k5_chain_1(A,B,C))) ) => k8_chain_1(A,B,C,D,E) = k8_chain_1(A,B,C,E,D) ) ), file(chain_1,k8_chain_1), []). 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(d12_chain_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & m2_subset_1(A,k1_numbers,k5_numbers) ) => ! [B] : ( m1_chain_1(B,A) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ! [D] : ( m2_subset_1(D,k1_zfmisc_1(k1_euclid(A)),k5_chain_1(A,B,C)) => k10_chain_1(A,B,C,D) = a_4_0_chain_1(A,B,C,D) ) ) ) ) ), file(chain_1,d12_chain_1), []). fof(d13_chain_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & m2_subset_1(A,k1_numbers,k5_numbers) ) => ! [B] : ( m1_chain_1(B,A) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(k5_chain_1(A,B,k1_nat_1(C,1)))) => k11_chain_1(A,B,C,D) = a_4_2_chain_1(A,B,C,D) ) ) ) ) ), file(chain_1,d13_chain_1), []). fof(d16_chain_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & m2_subset_1(A,k1_numbers,k5_numbers) ) => ! [B] : ( m1_chain_1(B,A) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ! [D] : ( m3_chain_1(D,A,B,k1_nat_1(C,1)) => k11_chain_1(A,B,C,D) = k6_chain_1(A,B,C) ) ) ) ) ), file(chain_1,d16_chain_1), []). 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(d2_xboole_0,axiom,( ! [A,B,C] : ( C = k2_xboole_0(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( r2_hidden(D,A) | r2_hidden(D,B) ) ) ) ), file(xboole_0,d2_xboole_0), []). 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(d4_chain_1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( ~ v1_xboole_0(B) & m1_finseq_2(B,k1_numbers) ) => ( B = k1_euclid(A) <=> ! [C] : ( r2_hidden(C,B) <=> ( v1_funct_1(C) & v1_funct_2(C,k2_finseq_1(A),k1_numbers) & m2_relset_1(C,k2_finseq_1(A),k1_numbers) ) ) ) ) ) ), file(chain_1,d4_chain_1), []). fof(d4_xboole_0,axiom,( ! [A,B,C] : ( C = k4_xboole_0(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( r2_hidden(D,A) & ~ r2_hidden(D,B) ) ) ) ), file(xboole_0,d4_xboole_0), []). fof(d5_tarski,axiom,( ! [A,B] : k4_tarski(A,B) = k2_tarski(k2_tarski(A,B),k1_tarski(A)) ), file(tarski,d5_tarski), []). fof(d7_chain_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & m2_subset_1(A,k1_numbers,k5_numbers) ) => ! [B] : ( m2_finseq_2(B,k1_numbers,k1_euclid(A)) => ! [C] : ( m2_finseq_2(C,k1_numbers,k1_euclid(A)) => k4_chain_1(A,B,C) = a_3_0_chain_1(A,B,C) ) ) ) ), file(chain_1,d7_chain_1), []). fof(d9_chain_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & m2_subset_1(A,k1_numbers,k5_numbers) ) => ! [B] : ( m1_chain_1(B,A) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => k6_chain_1(A,B,C) = k1_xboole_0 ) ) ) ), file(chain_1,d9_chain_1), []). fof(dt_k10_chain_1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k5_numbers) & m1_chain_1(B,A) & m1_subset_1(C,k5_numbers) & m1_subset_1(D,k5_chain_1(A,B,C)) ) => m1_subset_1(k10_chain_1(A,B,C,D),k1_zfmisc_1(k5_chain_1(A,B,k1_nat_1(C,1)))) ) ), file(chain_1,k10_chain_1), []). fof(dt_k11_chain_1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k5_numbers) & m1_chain_1(B,A) & m1_subset_1(C,k5_numbers) & m1_subset_1(D,k1_zfmisc_1(k5_chain_1(A,B,k1_nat_1(C,1)))) ) => m1_subset_1(k11_chain_1(A,B,C,D),k1_zfmisc_1(k5_chain_1(A,B,C))) ) ), file(chain_1,k11_chain_1), []). fof(dt_k1_card_1,axiom,( ! [A] : v1_card_1(k1_card_1(A)) ), file(card_1,k1_card_1), []). fof(dt_k1_domain_1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(C,A) & m1_subset_1(D,B) ) => m1_subset_1(k1_domain_1(A,B,C,D),k2_zfmisc_1(A,B)) ) ), file(domain_1,k1_domain_1), []). fof(dt_k1_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( ~ v1_xboole_0(k1_euclid(A)) & m1_finseq_2(k1_euclid(A),k1_numbers) ) ) ), file(euclid,k1_euclid), []). 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_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_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_k2_chain_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k5_numbers) & m1_subset_1(B,k1_euclid(A)) & m1_subset_1(C,k2_finseq_1(A)) ) => m1_subset_1(k2_chain_1(A,B,C),k1_numbers) ) ), file(chain_1,k2_chain_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_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k2_nat_1(A,B),k1_numbers,k5_numbers) ) ), file(nat_1,k2_nat_1), []). fof(dt_k2_tarski,axiom,( $true ), file(tarski,k2_tarski), []). fof(dt_k2_xboole_0,axiom,( $true ), file(xboole_0,k2_xboole_0), []). 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_chain_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k5_numbers) & m1_chain_1(B,A) & m1_subset_1(C,k2_finseq_1(A)) ) => ( v1_finset_1(k3_chain_1(A,B,C)) & ~ v1_realset1(k3_chain_1(A,B,C)) & m1_subset_1(k3_chain_1(A,B,C),k1_zfmisc_1(k1_numbers)) ) ) ), file(chain_1,k3_chain_1), []). fof(dt_k3_subset_1,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => m1_subset_1(k3_subset_1(A,B),k1_zfmisc_1(A)) ) ), file(subset_1,k3_subset_1), []). 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_card_1,axiom,( ! [A] : ( v1_finset_1(A) => m2_subset_1(k4_card_1(A),k1_numbers,k5_numbers) ) ), file(card_1,k4_card_1), []). fof(dt_k4_chain_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k5_numbers) & m1_subset_1(B,k1_euclid(A)) & m1_subset_1(C,k1_euclid(A)) ) => ( ~ v1_xboole_0(k4_chain_1(A,B,C)) & m1_subset_1(k4_chain_1(A,B,C),k1_zfmisc_1(k1_euclid(A))) ) ) ), file(chain_1,k4_chain_1), []). fof(dt_k4_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(k4_subset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(subset_1,k4_subset_1), []). fof(dt_k4_tarski,axiom,( $true ), file(tarski,k4_tarski), []). fof(dt_k4_xboole_0,axiom,( $true ), file(xboole_0,k4_xboole_0), []). fof(dt_k5_chain_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k5_numbers) & m1_chain_1(B,A) & m1_subset_1(C,k5_numbers) ) => ( ~ v1_xboole_0(k5_chain_1(A,B,C)) & v1_finset_1(k5_chain_1(A,B,C)) & m1_subset_1(k5_chain_1(A,B,C),k1_zfmisc_1(k1_zfmisc_1(k1_euclid(A)))) ) ) ), file(chain_1,k5_chain_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_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_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_k5_xboole_0,axiom,( $true ), file(xboole_0,k5_xboole_0), []). fof(dt_k6_chain_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k5_numbers) & m1_chain_1(B,A) & m1_subset_1(C,k5_numbers) ) => m1_subset_1(k6_chain_1(A,B,C),k1_zfmisc_1(k5_chain_1(A,B,C))) ) ), file(chain_1,k6_chain_1), []). fof(dt_k6_domain_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => m1_subset_1(k6_domain_1(A,B),k1_zfmisc_1(A)) ) ), file(domain_1,k6_domain_1), []). fof(dt_k6_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(k6_subset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(subset_1,k6_subset_1), []). fof(dt_k6_xcmplx_0,axiom,( $true ), file(xcmplx_0,k6_xcmplx_0), []). fof(dt_k8_chain_1,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k5_numbers) & m1_chain_1(B,A) & m1_subset_1(C,k5_numbers) & m1_subset_1(D,k1_zfmisc_1(k5_chain_1(A,B,C))) & m1_subset_1(E,k1_zfmisc_1(k5_chain_1(A,B,C))) ) => m1_subset_1(k8_chain_1(A,B,C,D,E),k1_zfmisc_1(k5_chain_1(A,B,C))) ) ), file(chain_1,k8_chain_1), []). fof(dt_k8_funct_2,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & m1_subset_1(D,A) ) => m1_subset_1(k8_funct_2(A,B,C,D),B) ) ), file(funct_2,k8_funct_2), []). fof(dt_k9_chain_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k5_numbers) & m1_chain_1(B,A) ) => m2_subset_1(k9_chain_1(A,B),k1_zfmisc_1(k1_euclid(A)),k5_chain_1(A,B,A)) ) ), file(chain_1,k9_chain_1), []). fof(dt_m1_chain_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k5_numbers) ) => ! [B] : ( m1_chain_1(B,A) => ( v1_funct_1(B) & v1_funct_2(B,k2_finseq_1(A),k1_zfmisc_1(k1_numbers)) & m2_relset_1(B,k2_finseq_1(A),k1_zfmisc_1(k1_numbers)) ) ) ) ), file(chain_1,m1_chain_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_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_chain_1,axiom,( ! [A] : ( ( v1_finset_1(A) & ~ v1_realset1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) ) => ! [B] : ( m2_chain_1(B,A) => m1_subset_1(B,k2_zfmisc_1(k1_numbers,k1_numbers)) ) ) ), file(chain_1,m2_chain_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_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_chain_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k5_numbers) & m1_chain_1(B,A) & m1_subset_1(C,k5_numbers) ) => ! [D] : ( m3_chain_1(D,A,B,C) => m1_subset_1(D,k1_zfmisc_1(k5_chain_1(A,B,C))) ) ) ), file(chain_1,m3_chain_1), []). fof(existence_m1_chain_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k5_numbers) ) => ? [B] : m1_chain_1(B,A) ) ), file(chain_1,m1_chain_1), []). 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_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_chain_1,axiom,( ! [A] : ( ( v1_finset_1(A) & ~ v1_realset1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) ) => ? [B] : m2_chain_1(B,A) ) ), file(chain_1,m2_chain_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_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_chain_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k5_numbers) & m1_chain_1(B,A) & m1_subset_1(C,k5_numbers) ) => ? [D] : m3_chain_1(D,A,B,C) ) ), file(chain_1,m3_chain_1), []). fof(fc10_abian,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_abian(A) & v1_int_1(B) & ~ v1_abian(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)) & ~ v1_abian(k2_xcmplx_0(A,B)) ) ) ), file(abian,fc10_abian), []). 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(fc11_abian,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_abian(A) & v1_int_1(B) & ~ v1_abian(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)) & ~ v1_abian(k2_xcmplx_0(B,A)) ) ) ), file(abian,fc11_abian), []). 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(fc12_abian,axiom,( ! [A,B] : ( ( v1_int_1(A) & ~ v1_abian(A) & v1_int_1(B) & ~ v1_abian(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)) & v1_abian(k2_xcmplx_0(A,B)) ) ) ), file(abian,fc12_abian), []). fof(fc12_finset_1,axiom,( ! [A,B] : ( v1_finset_1(A) => v1_finset_1(k4_xboole_0(A,B)) ) ), file(finset_1,fc12_finset_1), []). 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(fc13_abian,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_abian(A) & v1_int_1(B) & ~ v1_abian(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v1_int_1(k6_xcmplx_0(A,B)) & ~ v1_abian(k6_xcmplx_0(A,B)) ) ) ), file(abian,fc13_abian), []). 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_abian,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_abian(A) & v1_int_1(B) & ~ v1_abian(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & v1_int_1(k6_xcmplx_0(B,A)) & ~ v1_abian(k6_xcmplx_0(B,A)) ) ) ), file(abian,fc14_abian), []). 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(fc15_abian,axiom,( ! [A,B] : ( ( v1_int_1(A) & ~ v1_abian(A) & v1_int_1(B) & ~ v1_abian(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v1_int_1(k6_xcmplx_0(A,B)) & v1_abian(k6_xcmplx_0(A,B)) ) ) ), file(abian,fc15_abian), []). 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_abian,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ( ~ v1_xboole_0(k1_zfmisc_1(A)) & ~ v2_setfam_1(k1_zfmisc_1(A)) ) ) ), file(abian,fc17_abian), []). fof(fc17_finset_1,axiom,( ! [A,B] : ( ( v1_finset_1(A) & v1_finset_1(B) ) => v1_finset_1(k5_xboole_0(A,B)) ) ), file(finset_1,fc17_finset_1), []). fof(fc1_abian,axiom,( ! [A] : ( v1_int_1(A) => ( v1_xcmplx_0(k3_xcmplx_0(2,A)) & v1_xreal_0(k3_xcmplx_0(2,A)) & v1_int_1(k3_xcmplx_0(2,A)) & v1_abian(k3_xcmplx_0(2,A)) ) ) ), file(abian,fc1_abian), []). fof(fc1_chain_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k5_numbers) ) => ( ~ v1_xboole_0(k1_finseq_1(A)) & v1_finset_1(k1_finseq_1(A)) ) ) ), file(chain_1,fc1_chain_1), []). fof(fc1_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => v1_finset_1(k1_finseq_1(A)) ) ), file(finseq_1,fc1_finseq_1), []). 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_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_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ), file(subset_1,fc1_subset_1), []). fof(fc22_membered,axiom,( ! [A,B] : ( ( v1_membered(A) & v1_membered(B) ) => v1_membered(k2_xboole_0(A,B)) ) ), file(membered,fc22_membered), []). fof(fc23_membered,axiom,( ! [A,B] : ( ( v2_membered(A) & v2_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc23_membered), []). fof(fc24_membered,axiom,( ! [A,B] : ( ( v3_membered(A) & v3_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) & v3_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc24_membered), []). fof(fc25_membered,axiom,( ! [A,B] : ( ( v4_membered(A) & v4_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) & v3_membered(k2_xboole_0(A,B)) & v4_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc25_membered), []). fof(fc26_membered,axiom,( ! [A,B] : ( ( v5_membered(A) & v5_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) & v3_membered(k2_xboole_0(A,B)) & v4_membered(k2_xboole_0(A,B)) & v5_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc26_membered), []). 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_abian,axiom,( ! [A] : ( ( v1_int_1(A) & v1_abian(A) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,1)) & v1_xreal_0(k2_xcmplx_0(A,1)) & v1_int_1(k2_xcmplx_0(A,1)) & ~ v1_abian(k2_xcmplx_0(A,1)) ) ) ), file(abian,fc2_abian), []). fof(fc2_afinsq_1,axiom, ( v1_xboole_0(k1_xboole_0) & v1_relat_1(k1_xboole_0) & v3_relat_1(k1_xboole_0) & v1_funct_1(k1_xboole_0) & v2_funct_1(k1_xboole_0) & v1_ordinal1(k1_xboole_0) & v2_ordinal1(k1_xboole_0) & v3_ordinal1(k1_xboole_0) & v5_ordinal1(k1_xboole_0) & v4_ordinal2(k1_xboole_0) & v1_xcmplx_0(k1_xboole_0) & v1_finset_1(k1_xboole_0) & v1_finseq_1(k1_xboole_0) & v1_funcop_1(k1_xboole_0) & v1_funct_7(k1_xboole_0) & v1_xreal_0(k1_xboole_0) & ~ v2_xreal_0(k1_xboole_0) & ~ v3_xreal_0(k1_xboole_0) ), file(afinsq_1,fc2_afinsq_1), []). fof(fc2_chain_1,axiom,( ! [A,B] : ( ~ v1_realset1(A) => ( ~ v1_xboole_0(k2_xboole_0(A,B)) & ~ v1_realset1(k2_xboole_0(A,B)) ) ) ), file(chain_1,fc2_chain_1), []). fof(fc2_finseq_1,axiom, ( v1_xboole_0(k1_xboole_0) & v1_relat_1(k1_xboole_0) & v3_relat_1(k1_xboole_0) & v1_funct_1(k1_xboole_0) & v2_funct_1(k1_xboole_0) & v1_finset_1(k1_xboole_0) & v1_finseq_1(k1_xboole_0) ), file(finseq_1,fc2_finseq_1), []). 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_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_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_tarski(A)) ), file(subset_1,fc2_subset_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(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(fc37_membered,axiom,( ! [A,B] : ( v1_membered(A) => v1_membered(k4_xboole_0(A,B)) ) ), file(membered,fc37_membered), []). fof(fc38_membered,axiom,( ! [A,B] : ( v2_membered(A) => ( v1_membered(k4_xboole_0(A,B)) & v2_membered(k4_xboole_0(A,B)) ) ) ), file(membered,fc38_membered), []). fof(fc39_membered,axiom,( ! [A,B] : ( v3_membered(A) => ( v1_membered(k4_xboole_0(A,B)) & v2_membered(k4_xboole_0(A,B)) & v3_membered(k4_xboole_0(A,B)) ) ) ), file(membered,fc39_membered), []). fof(fc3_abian,axiom,( ! [A] : ( ( v1_int_1(A) & ~ v1_abian(A) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,1)) & v1_xreal_0(k2_xcmplx_0(A,1)) & v1_int_1(k2_xcmplx_0(A,1)) & v1_abian(k2_xcmplx_0(A,1)) ) ) ), file(abian,fc3_abian), []). fof(fc3_chain_1,axiom,( ! [A,B] : ( ~ v1_realset1(A) => ( ~ v1_xboole_0(k2_xboole_0(B,A)) & ~ v1_realset1(k2_xboole_0(B,A)) ) ) ), file(chain_1,fc3_chain_1), []). 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_realset1,axiom,( ! [A] : ( ~ v1_xboole_0(k1_tarski(A)) & v1_finset_1(k1_tarski(A)) & v1_realset1(k1_tarski(A)) ) ), file(realset1,fc3_realset1), []). fof(fc3_subset_1,axiom,( ! [A,B] : ~ v1_xboole_0(k2_tarski(A,B)) ), file(subset_1,fc3_subset_1), []). fof(fc40_membered,axiom,( ! [A,B] : ( v4_membered(A) => ( v1_membered(k4_xboole_0(A,B)) & v2_membered(k4_xboole_0(A,B)) & v3_membered(k4_xboole_0(A,B)) & v4_membered(k4_xboole_0(A,B)) ) ) ), file(membered,fc40_membered), []). fof(fc41_membered,axiom,( ! [A,B] : ( v5_membered(A) => ( v1_membered(k4_xboole_0(A,B)) & v2_membered(k4_xboole_0(A,B)) & v3_membered(k4_xboole_0(A,B)) & v4_membered(k4_xboole_0(A,B)) & v5_membered(k4_xboole_0(A,B)) ) ) ), file(membered,fc41_membered), []). fof(fc42_membered,axiom,( ! [A,B] : ( ( v1_membered(A) & v1_membered(B) ) => v1_membered(k5_xboole_0(A,B)) ) ), file(membered,fc42_membered), []). fof(fc43_membered,axiom,( ! [A,B] : ( ( v2_membered(A) & v2_membered(B) ) => ( v1_membered(k5_xboole_0(A,B)) & v2_membered(k5_xboole_0(A,B)) ) ) ), file(membered,fc43_membered), []). fof(fc44_membered,axiom,( ! [A,B] : ( ( v3_membered(A) & v3_membered(B) ) => ( v1_membered(k5_xboole_0(A,B)) & v2_membered(k5_xboole_0(A,B)) & v3_membered(k5_xboole_0(A,B)) ) ) ), file(membered,fc44_membered), []). fof(fc45_membered,axiom,( ! [A,B] : ( ( v4_membered(A) & v4_membered(B) ) => ( v1_membered(k5_xboole_0(A,B)) & v2_membered(k5_xboole_0(A,B)) & v3_membered(k5_xboole_0(A,B)) & v4_membered(k5_xboole_0(A,B)) ) ) ), file(membered,fc45_membered), []). fof(fc46_membered,axiom,( ! [A,B] : ( ( v5_membered(A) & v5_membered(B) ) => ( v1_membered(k5_xboole_0(A,B)) & v2_membered(k5_xboole_0(A,B)) & v3_membered(k5_xboole_0(A,B)) & v4_membered(k5_xboole_0(A,B)) & v5_membered(k5_xboole_0(A,B)) ) ) ), file(membered,fc46_membered), []). fof(fc4_abian,axiom,( ! [A] : ( ( v1_int_1(A) & v1_abian(A) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,1)) & v1_xreal_0(k6_xcmplx_0(A,1)) & v1_int_1(k6_xcmplx_0(A,1)) & ~ v1_abian(k6_xcmplx_0(A,1)) ) ) ), file(abian,fc4_abian), []). fof(fc4_chain_1,axiom, ( ~ v1_xboole_0(k1_numbers) & ~ v1_realset1(k1_numbers) & v1_membered(k1_numbers) & v2_membered(k1_numbers) ), file(chain_1,fc4_chain_1), []). 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_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ), file(subset_1,fc4_subset_1), []). fof(fc5_abian,axiom,( ! [A] : ( ( v1_int_1(A) & ~ v1_abian(A) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,1)) & v1_xreal_0(k6_xcmplx_0(A,1)) & v1_int_1(k6_xcmplx_0(A,1)) & v1_abian(k6_xcmplx_0(A,1)) ) ) ), file(abian,fc5_abian), []). 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(fc6_abian,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_abian(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)) & v1_abian(k3_xcmplx_0(A,B)) ) ) ), file(abian,fc6_abian), []). 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_abian,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_abian(A) & 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)) & v1_abian(k3_xcmplx_0(B,A)) ) ) ), file(abian,fc7_abian), []). fof(fc7_membered,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_membered(k1_tarski(A)) ) ), file(membered,fc7_membered), []). fof(fc8_abian,axiom,( ! [A,B] : ( ( v1_int_1(A) & ~ v1_abian(A) & v1_int_1(B) & ~ v1_abian(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)) & ~ v1_abian(k3_xcmplx_0(A,B)) ) ) ), file(abian,fc8_abian), []). fof(fc8_membered,axiom,( ! [A] : ( v1_xreal_0(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) ) ) ), file(membered,fc8_membered), []). fof(fc9_abian,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_abian(A) & v1_int_1(B) & v1_abian(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)) & v1_abian(k2_xcmplx_0(A,B)) ) ) ), file(abian,fc9_abian), []). fof(fc9_finset_1,axiom,( ! [A,B] : ( ( v1_finset_1(A) & v1_finset_1(B) ) => v1_finset_1(k2_xboole_0(A,B)) ) ), file(finset_1,fc9_finset_1), []). 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(fraenkel_a_3_0_chain_1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(B) & m2_subset_1(B,k1_numbers,k5_numbers) & m2_finseq_2(C,k1_numbers,k1_euclid(B)) & m2_finseq_2(D,k1_numbers,k1_euclid(B)) ) => ( r2_hidden(A,a_3_0_chain_1(B,C,D)) <=> ? [E] : ( m2_finseq_2(E,k1_numbers,k1_euclid(B)) & A = E & ~ ( ~ ! [F] : ( m1_subset_1(F,k2_finseq_1(B)) => ( r1_xreal_0(k2_chain_1(B,C,F),k2_chain_1(B,E,F)) & r1_xreal_0(k2_chain_1(B,E,F),k2_chain_1(B,D,F)) ) ) & ! [F] : ( m1_subset_1(F,k2_finseq_1(B)) => ~ ( ~ r1_xreal_0(k2_chain_1(B,C,F),k2_chain_1(B,D,F)) & ( r1_xreal_0(k2_chain_1(B,E,F),k2_chain_1(B,D,F)) | r1_xreal_0(k2_chain_1(B,C,F),k2_chain_1(B,E,F)) ) ) ) ) ) ) ) ), file(chain_1,a_3_0_chain_1), []). fof(fraenkel_a_4_0_chain_1,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(B) & m2_subset_1(B,k1_numbers,k5_numbers) & m1_chain_1(C,B) & m2_subset_1(D,k1_numbers,k5_numbers) & m2_subset_1(E,k1_zfmisc_1(k1_euclid(B)),k5_chain_1(B,C,D)) ) => ( r2_hidden(A,a_4_0_chain_1(B,C,D,E)) <=> ? [F] : ( m2_subset_1(F,k1_zfmisc_1(k1_euclid(B)),k5_chain_1(B,C,k1_nat_1(D,1))) & A = F & r1_tarski(E,F) ) ) ) ), file(chain_1,a_4_0_chain_1), []). fof(fraenkel_a_4_2_chain_1,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(B) & m2_subset_1(B,k1_numbers,k5_numbers) & m1_chain_1(C,B) & m2_subset_1(D,k1_numbers,k5_numbers) & m1_subset_1(E,k1_zfmisc_1(k5_chain_1(B,C,k1_nat_1(D,1)))) ) => ( r2_hidden(A,a_4_2_chain_1(B,C,D,E)) <=> ? [F] : ( m2_subset_1(F,k1_zfmisc_1(k1_euclid(B)),k5_chain_1(B,C,D)) & A = F & r1_xreal_0(k1_nat_1(D,1),B) & ~ v1_abian(k4_card_1(k5_subset_1(k5_chain_1(B,C,k1_nat_1(D,1)),k10_chain_1(B,C,D,F),E))) ) ) ) ), file(chain_1,a_4_2_chain_1), []). fof(idempotence_k2_xboole_0,axiom,( ! [A,B] : k2_xboole_0(A,A) = A ), file(xboole_0,k2_xboole_0), []). fof(idempotence_k3_xboole_0,axiom,( ! [A,B] : k3_xboole_0(A,A) = A ), file(xboole_0,k3_xboole_0), []). fof(idempotence_k4_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k4_subset_1(A,B,B) = B ) ), file(subset_1,k4_subset_1), []). 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(involutiveness_k3_subset_1,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => k3_subset_1(A,k3_subset_1(A,B)) = B ) ), file(subset_1,k3_subset_1), []). fof(rc1_abian,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_abian(A) ) ), file(abian,rc1_abian), []). fof(rc1_afinsq_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_finset_1(A) ) ), file(afinsq_1,rc1_afinsq_1), []). fof(rc1_chain_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_finset_1(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_int_1(A) & v1_rat_1(A) ) ), file(chain_1,rc1_chain_1), []). fof(rc1_finseq_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc1_finseq_1), []). fof(rc1_finseq_5,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) ) ), file(finseq_5,rc1_finseq_5), []). fof(rc1_finset_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) ) ), file(finset_1,rc1_finset_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_realset1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_realset1(A) ) ), file(realset1,rc1_realset1), []). fof(rc1_subset_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) ) ) ), file(subset_1,rc1_subset_1), []). fof(rc2_abian,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_abian(A) ) ), file(abian,rc2_abian), []). fof(rc2_chain_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) & ~ v1_realset1(A) ) ), file(chain_1,rc2_chain_1), []). fof(rc2_finseq_1,axiom,( ! [A] : ? [B] : ( m1_relset_1(B,k5_numbers,A) & v1_relat_1(B) & v1_funct_1(B) & v1_finset_1(B) & v1_finseq_1(B) ) ), file(finseq_1,rc2_finseq_1), []). fof(rc2_finseq_5,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_finseq_1(B,A) & ~ v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v2_funct_1(B) & v1_finset_1(B) & v1_finseq_1(B) ) ) ), file(finseq_5,rc2_finseq_5), []). 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_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_realset1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & ~ v1_realset1(A) ) ), file(realset1,rc2_realset1), []). fof(rc2_subset_1,axiom,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) ) ), file(subset_1,rc2_subset_1), []). fof(rc3_abian,axiom,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) & v1_int_1(A) & v1_abian(A) ) ), file(abian,rc3_abian), []). fof(rc3_afinsq_1,axiom,( ! [A] : ? [B] : ( m1_relset_1(B,k5_numbers,A) & v1_relat_1(B) & v1_funct_1(B) & v5_ordinal1(B) & v1_finset_1(B) ) ), file(afinsq_1,rc3_afinsq_1), []). fof(rc3_chain_1,axiom,( ! [A] : ( ~ v1_realset1(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & v1_finset_1(B) & ~ v1_realset1(B) ) ) ), file(chain_1,rc3_chain_1), []). fof(rc3_finseq_1,axiom,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc3_finseq_1), []). 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_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(rc4_abian,axiom,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) & v1_int_1(A) & ~ v1_abian(A) ) ), file(abian,rc4_abian), []). fof(rc4_afinsq_1,axiom,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v5_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_finset_1(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(afinsq_1,rc4_afinsq_1), []). fof(rc4_finseq_1,axiom,( ! [A] : ? [B] : ( m1_finseq_1(B,A) & v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v2_funct_1(B) & v1_finset_1(B) & v1_finseq_1(B) ) ), file(finseq_1,rc4_finseq_1), []). 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(rc6_finseq_1,axiom,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) ) ), file(finseq_1,rc6_finseq_1), []). fof(rc7_finseq_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) ), file(finseq_1,rc7_finseq_1), []). fof(rc8_finseq_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc8_finseq_1), []). fof(redefinition_k1_domain_1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(C,A) & m1_subset_1(D,B) ) => k1_domain_1(A,B,C,D) = k4_tarski(C,D) ) ), file(domain_1,k1_domain_1), []). 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_k2_chain_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k5_numbers) & m1_subset_1(B,k1_euclid(A)) & m1_subset_1(C,k2_finseq_1(A)) ) => k2_chain_1(A,B,C) = k1_funct_1(B,C) ) ), file(chain_1,k2_chain_1), []). 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_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k2_nat_1(A,B) = k3_xcmplx_0(A,B) ) ), file(nat_1,k2_nat_1), []). fof(redefinition_k3_chain_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k5_numbers) & m1_chain_1(B,A) & m1_subset_1(C,k2_finseq_1(A)) ) => k3_chain_1(A,B,C) = k1_funct_1(B,C) ) ), file(chain_1,k3_chain_1), []). fof(redefinition_k4_card_1,axiom,( ! [A] : ( v1_finset_1(A) => k4_card_1(A) = k1_card_1(A) ) ), file(card_1,k4_card_1), []). fof(redefinition_k4_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k4_subset_1(A,B,C) = k2_xboole_0(B,C) ) ), file(subset_1,k4_subset_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_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_domain_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => k6_domain_1(A,B) = k1_tarski(B) ) ), file(domain_1,k6_domain_1), []). fof(redefinition_k6_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k6_subset_1(A,B,C) = k4_xboole_0(B,C) ) ), file(subset_1,k6_subset_1), []). fof(redefinition_k8_chain_1,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k5_numbers) & m1_chain_1(B,A) & m1_subset_1(C,k5_numbers) & m1_subset_1(D,k1_zfmisc_1(k5_chain_1(A,B,C))) & m1_subset_1(E,k1_zfmisc_1(k5_chain_1(A,B,C))) ) => k8_chain_1(A,B,C,D,E) = k5_xboole_0(D,E) ) ), file(chain_1,k8_chain_1), []). fof(redefinition_k8_funct_2,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & m1_subset_1(D,A) ) => k8_funct_2(A,B,C,D) = k1_funct_1(C,D) ) ), file(funct_2,k8_funct_2), []). 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(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(s3_funct_2__e11_92_1_1_1_1_2__chain_1,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(A) & m2_subset_1(A,k1_numbers,k5_numbers) & m2_finseq_2(B,k1_numbers,k1_euclid(A)) & m2_finseq_2(C,k1_numbers,k1_euclid(A)) & m1_subset_1(D,k1_zfmisc_1(k2_finseq_1(A))) & m1_subset_1(E,k2_finseq_1(A)) ) => ( ! [F] : ( m1_subset_1(F,k2_finseq_1(A)) => ? [G] : ( m1_subset_1(G,k1_numbers) & ( r2_hidden(F,k4_subset_1(k2_finseq_1(A),D,k6_domain_1(k2_finseq_1(A),E))) => G = k2_chain_1(A,B,F) ) & ( ~ r2_hidden(F,k4_subset_1(k2_finseq_1(A),D,k6_domain_1(k2_finseq_1(A),E))) => G = k2_chain_1(A,C,F) ) ) ) => ? [F] : ( v1_funct_1(F) & v1_funct_2(F,k2_finseq_1(A),k1_numbers) & m2_relset_1(F,k2_finseq_1(A),k1_numbers) & ! [G] : ( m1_subset_1(G,k2_finseq_1(A)) => ( ( r2_hidden(G,k4_subset_1(k2_finseq_1(A),D,k6_domain_1(k2_finseq_1(A),E))) => k8_funct_2(k2_finseq_1(A),k1_numbers,F,G) = k2_chain_1(A,B,G) ) & ( ~ r2_hidden(G,k4_subset_1(k2_finseq_1(A),D,k6_domain_1(k2_finseq_1(A),E))) => k8_funct_2(k2_finseq_1(A),k1_numbers,F,G) = k2_chain_1(A,C,G) ) ) ) ) ) ) ), file(chain_1,s3_funct_2__e11_92_1_1_1_1_2__chain_1), []). fof(s3_funct_2__e14_92_1_1_1_1_2__chain_1,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(A) & m2_subset_1(A,k1_numbers,k5_numbers) & m2_finseq_2(B,k1_numbers,k1_euclid(A)) & m2_finseq_2(C,k1_numbers,k1_euclid(A)) & m1_subset_1(D,k1_zfmisc_1(k2_finseq_1(A))) & m1_subset_1(E,k2_finseq_1(A)) ) => ( ! [F] : ( m1_subset_1(F,k2_finseq_1(A)) => ? [G] : ( m1_subset_1(G,k1_numbers) & ( r2_hidden(F,k4_subset_1(k2_finseq_1(A),D,k6_domain_1(k2_finseq_1(A),E))) => G = k2_chain_1(A,B,F) ) & ( ~ r2_hidden(F,k4_subset_1(k2_finseq_1(A),D,k6_domain_1(k2_finseq_1(A),E))) => G = k2_chain_1(A,C,F) ) ) ) => ? [F] : ( v1_funct_1(F) & v1_funct_2(F,k2_finseq_1(A),k1_numbers) & m2_relset_1(F,k2_finseq_1(A),k1_numbers) & ! [G] : ( m1_subset_1(G,k2_finseq_1(A)) => ( ( r2_hidden(G,k4_subset_1(k2_finseq_1(A),D,k6_domain_1(k2_finseq_1(A),E))) => k8_funct_2(k2_finseq_1(A),k1_numbers,F,G) = k2_chain_1(A,B,G) ) & ( ~ r2_hidden(G,k4_subset_1(k2_finseq_1(A),D,k6_domain_1(k2_finseq_1(A),E))) => k8_funct_2(k2_finseq_1(A),k1_numbers,F,G) = k2_chain_1(A,C,G) ) ) ) ) ) ) ), file(chain_1,s3_funct_2__e14_92_1_1_1_1_2__chain_1), []). fof(s3_funct_2__e26_92_1_1_1_1_2__chain_1,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(A) & m2_subset_1(A,k1_numbers,k5_numbers) & m2_finseq_2(B,k1_numbers,k1_euclid(A)) & m2_finseq_2(C,k1_numbers,k1_euclid(A)) & m1_subset_1(D,k1_zfmisc_1(k2_finseq_1(A))) & m1_subset_1(E,k2_finseq_1(A)) ) => ( ! [F] : ( m1_subset_1(F,k2_finseq_1(A)) => ? [G] : ( m1_subset_1(G,k1_numbers) & ( r2_hidden(F,k4_subset_1(k2_finseq_1(A),D,k6_domain_1(k2_finseq_1(A),E))) => G = k2_chain_1(A,B,F) ) & ( ~ r2_hidden(F,k4_subset_1(k2_finseq_1(A),D,k6_domain_1(k2_finseq_1(A),E))) => G = k2_chain_1(A,C,F) ) ) ) => ? [F] : ( v1_funct_1(F) & v1_funct_2(F,k2_finseq_1(A),k1_numbers) & m2_relset_1(F,k2_finseq_1(A),k1_numbers) & ! [G] : ( m1_subset_1(G,k2_finseq_1(A)) => ( ( r2_hidden(G,k4_subset_1(k2_finseq_1(A),D,k6_domain_1(k2_finseq_1(A),E))) => k8_funct_2(k2_finseq_1(A),k1_numbers,F,G) = k2_chain_1(A,B,G) ) & ( ~ r2_hidden(G,k4_subset_1(k2_finseq_1(A),D,k6_domain_1(k2_finseq_1(A),E))) => k8_funct_2(k2_finseq_1(A),k1_numbers,F,G) = k2_chain_1(A,C,G) ) ) ) ) ) ) ), file(chain_1,s3_funct_2__e26_92_1_1_1_1_2__chain_1), []). fof(s3_funct_2__e29_92_1_1_1_1_2__chain_1,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(A) & m2_subset_1(A,k1_numbers,k5_numbers) & m2_finseq_2(B,k1_numbers,k1_euclid(A)) & m2_finseq_2(C,k1_numbers,k1_euclid(A)) & m1_subset_1(D,k1_zfmisc_1(k2_finseq_1(A))) & m1_subset_1(E,k2_finseq_1(A)) ) => ( ! [F] : ( m1_subset_1(F,k2_finseq_1(A)) => ? [G] : ( m1_subset_1(G,k1_numbers) & ( r2_hidden(F,k4_subset_1(k2_finseq_1(A),D,k6_domain_1(k2_finseq_1(A),E))) => G = k2_chain_1(A,B,F) ) & ( ~ r2_hidden(F,k4_subset_1(k2_finseq_1(A),D,k6_domain_1(k2_finseq_1(A),E))) => G = k2_chain_1(A,C,F) ) ) ) => ? [F] : ( v1_funct_1(F) & v1_funct_2(F,k2_finseq_1(A),k1_numbers) & m2_relset_1(F,k2_finseq_1(A),k1_numbers) & ! [G] : ( m1_subset_1(G,k2_finseq_1(A)) => ( ( r2_hidden(G,k4_subset_1(k2_finseq_1(A),D,k6_domain_1(k2_finseq_1(A),E))) => k8_funct_2(k2_finseq_1(A),k1_numbers,F,G) = k2_chain_1(A,B,G) ) & ( ~ r2_hidden(G,k4_subset_1(k2_finseq_1(A),D,k6_domain_1(k2_finseq_1(A),E))) => k8_funct_2(k2_finseq_1(A),k1_numbers,F,G) = k2_chain_1(A,C,G) ) ) ) ) ) ) ), file(chain_1,s3_funct_2__e29_92_1_1_1_1_2__chain_1), []). fof(s4_chain_1__e10_92_1_1__chain_1,axiom,( ! [A,B,C,D] : ( ( m2_subset_1(A,k1_numbers,k5_numbers) & ~ v1_xboole_0(B) & m2_subset_1(B,k1_numbers,k5_numbers) & m1_chain_1(C,B) & m1_subset_1(D,k1_zfmisc_1(k5_chain_1(B,C,k1_nat_1(k1_nat_1(A,1),1)))) ) => ( ( k11_chain_1(B,C,A,k11_chain_1(B,C,k1_nat_1(A,1),k6_chain_1(B,C,k1_nat_1(k1_nat_1(A,1),1)))) = k6_chain_1(B,C,A) & ! [E] : ( m2_subset_1(E,k1_zfmisc_1(k1_euclid(B)),k5_chain_1(B,C,k1_nat_1(k1_nat_1(A,1),1))) => ( r2_hidden(E,D) => k11_chain_1(B,C,A,k11_chain_1(B,C,k1_nat_1(A,1),k6_domain_1(k5_chain_1(B,C,k1_nat_1(k1_nat_1(A,1),1)),E))) = k6_chain_1(B,C,A) ) ) & ! [E] : ( m1_subset_1(E,k1_zfmisc_1(k5_chain_1(B,C,k1_nat_1(k1_nat_1(A,1),1)))) => ! [F] : ( m1_subset_1(F,k1_zfmisc_1(k5_chain_1(B,C,k1_nat_1(k1_nat_1(A,1),1)))) => ( ( r1_tarski(E,D) & r1_tarski(F,D) & k11_chain_1(B,C,A,k11_chain_1(B,C,k1_nat_1(A,1),E)) = k6_chain_1(B,C,A) & k11_chain_1(B,C,A,k11_chain_1(B,C,k1_nat_1(A,1),F)) = k6_chain_1(B,C,A) ) => k11_chain_1(B,C,A,k11_chain_1(B,C,k1_nat_1(A,1),k8_chain_1(B,C,k1_nat_1(k1_nat_1(A,1),1),E,F))) = k6_chain_1(B,C,A) ) ) ) ) => k11_chain_1(B,C,A,k11_chain_1(B,C,k1_nat_1(A,1),D)) = k6_chain_1(B,C,A) ) ) ), file(chain_1,s4_chain_1__e10_92_1_1__chain_1), []). fof(s4_chain_1__e5_92_1_1_4_1_2__chain_1,axiom,( ! [A,B,C,D] : ( ( m2_subset_1(A,k1_numbers,k5_numbers) & ~ v1_xboole_0(B) & m2_subset_1(B,k1_numbers,k5_numbers) & m1_chain_1(C,B) & m2_subset_1(D,k1_zfmisc_1(k1_euclid(B)),k5_chain_1(B,C,k1_nat_1(k1_nat_1(A,1),1))) ) => ( ( k11_chain_1(B,C,A,k11_chain_1(B,C,k1_nat_1(A,1),k6_chain_1(B,C,k1_nat_1(k1_nat_1(A,1),1)))) = k6_chain_1(B,C,A) & ! [E] : ( m2_subset_1(E,k1_zfmisc_1(k1_euclid(B)),k5_chain_1(B,C,k1_nat_1(k1_nat_1(A,1),1))) => ( r2_hidden(E,k3_subset_1(k5_chain_1(B,C,k1_nat_1(k1_nat_1(A,1),1)),k6_domain_1(k5_chain_1(B,C,k1_nat_1(k1_nat_1(A,1),1)),D))) => k11_chain_1(B,C,A,k11_chain_1(B,C,k1_nat_1(A,1),k6_domain_1(k5_chain_1(B,C,k1_nat_1(k1_nat_1(A,1),1)),E))) = k6_chain_1(B,C,A) ) ) & ! [E] : ( m1_subset_1(E,k1_zfmisc_1(k5_chain_1(B,C,k1_nat_1(k1_nat_1(A,1),1)))) => ! [F] : ( m1_subset_1(F,k1_zfmisc_1(k5_chain_1(B,C,k1_nat_1(k1_nat_1(A,1),1)))) => ( ( r1_tarski(E,k3_subset_1(k5_chain_1(B,C,k1_nat_1(k1_nat_1(A,1),1)),k6_domain_1(k5_chain_1(B,C,k1_nat_1(k1_nat_1(A,1),1)),D))) & r1_tarski(F,k3_subset_1(k5_chain_1(B,C,k1_nat_1(k1_nat_1(A,1),1)),k6_domain_1(k5_chain_1(B,C,k1_nat_1(k1_nat_1(A,1),1)),D))) & k11_chain_1(B,C,A,k11_chain_1(B,C,k1_nat_1(A,1),E)) = k6_chain_1(B,C,A) & k11_chain_1(B,C,A,k11_chain_1(B,C,k1_nat_1(A,1),F)) = k6_chain_1(B,C,A) ) => k11_chain_1(B,C,A,k11_chain_1(B,C,k1_nat_1(A,1),k8_chain_1(B,C,k1_nat_1(k1_nat_1(A,1),1),E,F))) = k6_chain_1(B,C,A) ) ) ) ) => k11_chain_1(B,C,A,k11_chain_1(B,C,k1_nat_1(A,1),k3_subset_1(k5_chain_1(B,C,k1_nat_1(k1_nat_1(A,1),1)),k6_domain_1(k5_chain_1(B,C,k1_nat_1(k1_nat_1(A,1),1)),D)))) = k6_chain_1(B,C,A) ) ) ), file(chain_1,s4_chain_1__e5_92_1_1_4_1_2__chain_1), []). 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(t113_funct_2,axiom,( ! [A,B,C] : ( ( v1_funct_1(C) & v1_funct_2(C,A,B) & m2_relset_1(C,A,B) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,A,B) & m2_relset_1(D,A,B) ) => ( ! [E] : ( m1_subset_1(E,A) => k1_funct_1(C,E) = k1_funct_1(D,E) ) => C = D ) ) ) ), file(funct_2,t113_funct_2), []). fof(t12_xboole_1,axiom,( ! [A,B] : ( r1_tarski(A,B) => k2_xboole_0(A,B) = B ) ), file(xboole_1,t12_xboole_1), []). fof(t1_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k2_xcmplx_0(A,0) = A ) ), file(arithm,t1_arithm), []). fof(t1_boole,axiom,( ! [A] : k2_xboole_0(A,k1_xboole_0) = A ), file(boole,t1_boole), []). 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(t29_chain_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & m2_subset_1(A,k1_numbers,k5_numbers) ) => ! [B] : ( m2_finseq_2(B,k1_numbers,k1_euclid(A)) => ! [C] : ( m2_finseq_2(C,k1_numbers,k1_euclid(A)) => ! [D] : ( m2_finseq_2(D,k1_numbers,k1_euclid(A)) => ! [E] : ( m2_finseq_2(E,k1_numbers,k1_euclid(A)) => ( ! [F] : ( m1_subset_1(F,k2_finseq_1(A)) => r1_xreal_0(k2_chain_1(A,B,F),k2_chain_1(A,C,F)) ) => ( r1_tarski(k4_chain_1(A,D,E),k4_chain_1(A,B,C)) <=> ! [F] : ( m1_subset_1(F,k2_finseq_1(A)) => ( r1_xreal_0(k2_chain_1(A,B,F),k2_chain_1(A,D,F)) & r1_xreal_0(k2_chain_1(A,D,F),k2_chain_1(A,E,F)) & r1_xreal_0(k2_chain_1(A,E,F),k2_chain_1(A,C,F)) ) ) ) ) ) ) ) ) ) ), file(chain_1,t29_chain_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(t32_chain_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & m2_subset_1(A,k1_numbers,k5_numbers) ) => ! [B] : ( m2_finseq_2(B,k1_numbers,k1_euclid(A)) => ! [C] : ( m2_finseq_2(C,k1_numbers,k1_euclid(A)) => ! [D] : ( m2_finseq_2(D,k1_numbers,k1_euclid(A)) => ! [E] : ( m2_finseq_2(E,k1_numbers,k1_euclid(A)) => ( ( ! [F] : ( m1_subset_1(F,k2_finseq_1(A)) => r1_xreal_0(k2_chain_1(A,B,F),k2_chain_1(A,C,F)) ) | ! [F] : ( m1_subset_1(F,k2_finseq_1(A)) => ~ r1_xreal_0(k2_chain_1(A,B,F),k2_chain_1(A,C,F)) ) ) => ( k4_chain_1(A,B,C) = k4_chain_1(A,D,E) <=> ( B = D & C = E ) ) ) ) ) ) ) ) ), file(chain_1,t32_chain_1), []). fof(t33_chain_1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( ~ v1_xboole_0(B) & m2_subset_1(B,k1_numbers,k5_numbers) ) => ! [C] : ( m1_chain_1(C,B) => ( r1_xreal_0(A,B) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(k1_euclid(B))) => ( r2_hidden(D,k5_chain_1(B,C,A)) <=> ? [E] : ( m2_finseq_2(E,k1_numbers,k1_euclid(B)) & ? [F] : ( m2_finseq_2(F,k1_numbers,k1_euclid(B)) & D = k4_chain_1(B,E,F) & ~ ( ! [G] : ( m1_subset_1(G,k1_zfmisc_1(k2_finseq_1(B))) => ~ ( k4_card_1(G) = A & ! [H] : ( m1_subset_1(H,k2_finseq_1(B)) => ( ( r2_hidden(H,G) & ~ r1_xreal_0(k2_chain_1(B,F,H),k2_chain_1(B,E,H)) & m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(B,E,H),k2_chain_1(B,F,H)),k3_chain_1(B,C,H)) ) | ( ~ r2_hidden(H,G) & k2_chain_1(B,E,H) = k2_chain_1(B,F,H) & r2_hidden(k2_chain_1(B,E,H),k3_chain_1(B,C,H)) ) ) ) ) ) & ~ ( A = B & ! [G] : ( m1_subset_1(G,k2_finseq_1(B)) => ( ~ r1_xreal_0(k2_chain_1(B,E,G),k2_chain_1(B,F,G)) & m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(B,E,G),k2_chain_1(B,F,G)),k3_chain_1(B,C,G)) ) ) ) ) ) ) ) ) ) ) ) ) ), file(chain_1,t33_chain_1), []). fof(t34_chain_1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( ~ v1_xboole_0(B) & m2_subset_1(B,k1_numbers,k5_numbers) ) => ! [C] : ( m2_finseq_2(C,k1_numbers,k1_euclid(B)) => ! [D] : ( m2_finseq_2(D,k1_numbers,k1_euclid(B)) => ! [E] : ( m1_chain_1(E,B) => ( r1_xreal_0(A,B) => ( r2_hidden(k4_chain_1(B,C,D),k5_chain_1(B,E,A)) <=> ~ ( ! [F] : ( m1_subset_1(F,k1_zfmisc_1(k2_finseq_1(B))) => ~ ( k4_card_1(F) = A & ! [G] : ( m1_subset_1(G,k2_finseq_1(B)) => ( ( r2_hidden(G,F) & ~ r1_xreal_0(k2_chain_1(B,D,G),k2_chain_1(B,C,G)) & m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(B,C,G),k2_chain_1(B,D,G)),k3_chain_1(B,E,G)) ) | ( ~ r2_hidden(G,F) & k2_chain_1(B,C,G) = k2_chain_1(B,D,G) & r2_hidden(k2_chain_1(B,C,G),k3_chain_1(B,E,G)) ) ) ) ) ) & ~ ( A = B & ! [F] : ( m1_subset_1(F,k2_finseq_1(B)) => ( ~ r1_xreal_0(k2_chain_1(B,C,F),k2_chain_1(B,D,F)) & m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(B,C,F),k2_chain_1(B,D,F)),k3_chain_1(B,E,F)) ) ) ) ) ) ) ) ) ) ) ) ), file(chain_1,t34_chain_1), []). fof(t37_zfmisc_1,axiom,( ! [A,B] : ( r1_tarski(k1_tarski(A),B) <=> r2_hidden(A,B) ) ), file(zfmisc_1,t37_zfmisc_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_xboole_1,axiom,( ! [A,B] : k2_xboole_0(A,k4_xboole_0(B,A)) = k2_xboole_0(A,B) ), file(xboole_1,t39_xboole_1), []). fof(t3_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(1,A) = A ) ), file(arithm,t3_arithm), []). fof(t3_boole,axiom,( ! [A] : k4_xboole_0(A,k1_xboole_0) = A ), file(boole,t3_boole), []). 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(t48_chain_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & m2_subset_1(A,k1_numbers,k5_numbers) ) => ! [B] : ( m2_finseq_2(B,k1_numbers,k1_euclid(A)) => ! [C] : ( m2_finseq_2(C,k1_numbers,k1_euclid(A)) => ! [D] : ( m2_finseq_2(D,k1_numbers,k1_euclid(A)) => ! [E] : ( m2_finseq_2(E,k1_numbers,k1_euclid(A)) => ! [F] : ( m1_chain_1(F,A) => ! [G] : ( m1_subset_1(G,k1_zfmisc_1(k2_finseq_1(A))) => ! [H] : ( m1_subset_1(H,k1_zfmisc_1(k2_finseq_1(A))) => ( ( r1_tarski(k4_chain_1(A,B,C),k4_chain_1(A,D,E)) & ! [I] : ( m1_subset_1(I,k2_finseq_1(A)) => ( ( r2_hidden(I,G) & ~ r1_xreal_0(k2_chain_1(A,C,I),k2_chain_1(A,B,I)) & m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(A,B,I),k2_chain_1(A,C,I)),k3_chain_1(A,F,I)) ) | ( ~ r2_hidden(I,G) & k2_chain_1(A,B,I) = k2_chain_1(A,C,I) & r2_hidden(k2_chain_1(A,B,I),k3_chain_1(A,F,I)) ) ) ) & ! [I] : ( m1_subset_1(I,k2_finseq_1(A)) => ( ( r2_hidden(I,H) & ~ r1_xreal_0(k2_chain_1(A,E,I),k2_chain_1(A,D,I)) & m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(A,D,I),k2_chain_1(A,E,I)),k3_chain_1(A,F,I)) ) | ( ~ r2_hidden(I,H) & k2_chain_1(A,D,I) = k2_chain_1(A,E,I) & r2_hidden(k2_chain_1(A,D,I),k3_chain_1(A,F,I)) ) ) ) ) => ( r1_tarski(G,H) & ! [I] : ( m1_subset_1(I,k2_finseq_1(A)) => ( ~ ( ~ r2_hidden(I,G) & r2_hidden(I,H) ) => ( k2_chain_1(A,B,I) = k2_chain_1(A,D,I) & k2_chain_1(A,C,I) = k2_chain_1(A,E,I) ) ) ) & ! [I] : ( m1_subset_1(I,k2_finseq_1(A)) => ~ ( ~ r2_hidden(I,G) & r2_hidden(I,H) & ~ ( k2_chain_1(A,B,I) = k2_chain_1(A,D,I) & k2_chain_1(A,C,I) = k2_chain_1(A,D,I) ) & ~ ( k2_chain_1(A,B,I) = k2_chain_1(A,E,I) & k2_chain_1(A,C,I) = k2_chain_1(A,E,I) ) ) ) ) ) ) ) ) ) ) ) ) ) ), file(chain_1,t48_chain_1), []). fof(t4_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k6_xcmplx_0(A,0) = A ) ), file(arithm,t4_arithm), []). fof(t4_boole,axiom,( ! [A] : k4_xboole_0(k1_xboole_0,A) = k1_xboole_0 ), file(boole,t4_boole), []). 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(t50_chain_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & m2_subset_1(A,k1_numbers,k5_numbers) ) => ! [B] : ( m2_finseq_2(B,k1_numbers,k1_euclid(A)) => ! [C] : ( m2_finseq_2(C,k1_numbers,k1_euclid(A)) => ! [D] : ( m1_chain_1(D,A) => ( k4_chain_1(A,B,C) = k9_chain_1(A,D) <=> ! [E] : ( m1_subset_1(E,k2_finseq_1(A)) => ( ~ r1_xreal_0(k2_chain_1(A,B,E),k2_chain_1(A,C,E)) & m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(A,B,E),k2_chain_1(A,C,E)),k3_chain_1(A,D,E)) ) ) ) ) ) ) ) ), file(chain_1,t50_chain_1), []). fof(t51_chain_1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( ~ v1_xboole_0(B) & m2_subset_1(B,k1_numbers,k5_numbers) ) => ! [C] : ( m1_chain_1(C,B) => ! [D] : ( m2_subset_1(D,k1_zfmisc_1(k1_euclid(B)),k5_chain_1(B,C,A)) => ! [E] : ( m2_subset_1(E,k1_zfmisc_1(k1_euclid(B)),k5_chain_1(B,C,k1_nat_1(A,1))) => ( r2_hidden(E,k10_chain_1(B,C,A,D)) <=> r1_tarski(D,E) ) ) ) ) ) ) ), file(chain_1,t51_chain_1), []). fof(t52_chain_1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( ~ v1_xboole_0(B) & m2_subset_1(B,k1_numbers,k5_numbers) ) => ! [C] : ( m1_chain_1(C,B) => ! [D] : ( m2_subset_1(D,k1_zfmisc_1(k1_euclid(B)),k5_chain_1(B,C,A)) => ! [E] : ( m1_subset_1(E,k1_zfmisc_1(k5_chain_1(B,C,k1_nat_1(A,1)))) => ( r2_hidden(D,k11_chain_1(B,C,A,E)) <=> ( r1_xreal_0(k1_nat_1(A,1),B) & ~ v1_abian(k4_card_1(k5_subset_1(k5_chain_1(B,C,k1_nat_1(A,1)),k10_chain_1(B,C,A,D),E))) ) ) ) ) ) ) ) ), file(chain_1,t52_chain_1), []). fof(t53_card_2,axiom,( ! [A] : ( v1_finset_1(A) => ! [B] : ( v1_finset_1(B) => ( r1_xboole_0(A,B) => k4_card_1(k2_xboole_0(A,B)) = k1_nat_1(k4_card_1(A),k4_card_1(B)) ) ) ) ), file(card_2,t53_card_2), []). fof(t53_chain_1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( ~ v1_xboole_0(B) & m2_subset_1(B,k1_numbers,k5_numbers) ) => ! [C] : ( m1_chain_1(C,B) => ( ~ r1_xreal_0(k1_nat_1(A,1),B) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(k5_chain_1(B,C,k1_nat_1(A,1)))) => k11_chain_1(B,C,A,D) = k6_chain_1(B,C,A) ) ) ) ) ) ), file(chain_1,t53_chain_1), []). fof(t54_chain_1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( ~ v1_xboole_0(B) & m2_subset_1(B,k1_numbers,k5_numbers) ) => ! [C] : ( m1_chain_1(C,B) => ( r1_xreal_0(k1_nat_1(A,1),B) => ! [D] : ( m2_subset_1(D,k1_zfmisc_1(k1_euclid(B)),k5_chain_1(B,C,A)) => ! [E] : ( m2_subset_1(E,k1_zfmisc_1(k1_euclid(B)),k5_chain_1(B,C,k1_nat_1(A,1))) => ( r2_hidden(D,k11_chain_1(B,C,A,k6_domain_1(k5_chain_1(B,C,k1_nat_1(A,1)),E))) <=> r1_tarski(D,E) ) ) ) ) ) ) ) ), file(chain_1,t54_chain_1), []). fof(t54_subset_1,axiom,( ! [A,B,C] : ( m1_subset_1(C,k1_zfmisc_1(A)) => ~ ( r2_hidden(B,k3_subset_1(A,C)) & r2_hidden(B,C) ) ) ), file(subset_1,t54_subset_1), []). fof(t56_zfmisc_1,axiom,( ! [A,B] : ( ~ r2_hidden(A,B) => r1_xboole_0(k1_tarski(A),B) ) ), file(zfmisc_1,t56_zfmisc_1), []). fof(t5_boole,axiom,( ! [A] : k5_xboole_0(A,k1_xboole_0) = A ), file(boole,t5_boole), []). 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(t60_card_2,axiom,( ! [A] : ( k1_card_1(A) = 1 <=> ? [B] : A = k1_tarski(B) ) ), file(card_2,t60_card_2), []). fof(t61_chain_1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( ~ v1_xboole_0(B) & m2_subset_1(B,k1_numbers,k5_numbers) ) => ! [C] : ( m1_chain_1(C,B) => k11_chain_1(B,C,A,k6_chain_1(B,C,k1_nat_1(A,1))) = k6_chain_1(B,C,A) ) ) ) ), file(chain_1,t61_chain_1), []). fof(t63_card_2,axiom,( ! [A] : ( v1_finset_1(A) => ! [B] : ( v1_finset_1(B) => ( r1_tarski(B,A) => k4_card_1(k4_xboole_0(A,B)) = k5_real_1(k4_card_1(A),k4_card_1(B)) ) ) ) ), file(card_2,t63_card_2), []). fof(t63_chain_1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( ~ v1_xboole_0(B) & m2_subset_1(B,k1_numbers,k5_numbers) ) => ! [C] : ( m1_chain_1(C,B) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(k5_chain_1(B,C,k1_nat_1(A,1)))) => ! [E] : ( m1_subset_1(E,k1_zfmisc_1(k5_chain_1(B,C,k1_nat_1(A,1)))) => k11_chain_1(B,C,A,k8_chain_1(B,C,k1_nat_1(A,1),D,E)) = k8_chain_1(B,C,A,k11_chain_1(B,C,A,D),k11_chain_1(B,C,A,E)) ) ) ) ) ) ), file(chain_1,t63_chain_1), []). fof(t64_chain_1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_chain_1(B,k1_nat_1(A,1)) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(k5_chain_1(k1_nat_1(A,1),B,k1_nat_1(A,1)))) => k11_chain_1(k1_nat_1(A,1),B,A,k3_subset_1(k5_chain_1(k1_nat_1(A,1),B,k1_nat_1(A,1)),C)) = k11_chain_1(k1_nat_1(A,1),B,A,C) ) ) ) ), file(chain_1,t64_chain_1), []). fof(t6_boole,axiom,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), []). fof(t6_chain_1,axiom,( ! [A] : ( k1_card_1(A) = 2 <=> ? [B,C] : ( r2_hidden(B,A) & r2_hidden(C,A) & B != C & ! [D] : ~ ( r2_hidden(D,A) & D != B & D != C ) ) ) ), file(chain_1,t6_chain_1), []). 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(t78_card_1,axiom,( k4_card_1(k1_xboole_0) = 0 ), file(card_1,t78_card_1), []). fof(t79_card_1,axiom,( ! [A] : k4_card_1(k1_tarski(A)) = 1 ), file(card_1,t79_card_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(t7_xboole_1,axiom,( ! [A,B] : r1_tarski(A,k2_xboole_0(A,B)) ), file(xboole_1,t7_xboole_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_xboole_1,axiom,( ! [A,B,C] : ( ( r1_tarski(A,B) & r1_tarski(C,B) ) => r1_tarski(k2_xboole_0(A,C),B) ) ), file(xboole_1,t8_xboole_1), []).