% Mizar problem: t18_bintree2,bintree2,923,33 fof(t18_bintree2,conjecture,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_bintree2(A) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => k1_card_1(k2_trees_2(A,B)) = k3_series_1(2,B) ) ) ), inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc1_bintree1,cc1_bintree2,cc1_finseq_1,cc1_finset_1,cc1_margrel1,cc1_nat_1,cc1_relset_1,cc1_trees_9,cc1_xreal_0,cc2_finset_1,cc2_nat_1,cc2_xreal_0,cc3_nat_1,cc3_trees_9,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,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_k7_domain_1,connectedness_r1_xreal_0,d10_xboole_0,d11_finseq_1,d12_margrel1,d1_funct_2,d1_tarski,d1_xboole_0,d2_bintree2,d2_tarski,d2_xboole_0,d3_finseq_1,d3_tarski,d3_xboole_0,d4_euclid,d4_finseq_1,d6_trees_2,d7_xboole_0,d8_funct_1,dt_k12_finseq_1,dt_k13_binarith,dt_k13_finseq_1,dt_k1_card_1,dt_k1_euclid,dt_k1_finseq_1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_finseq_1,dt_k2_finseq_2,dt_k2_funct_2,dt_k2_nat_1,dt_k2_relat_1,dt_k2_tarski,dt_k2_trees_2,dt_k2_xboole_0,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k3_finseq_1,dt_k3_finseq_2,dt_k3_power,dt_k3_series_1,dt_k3_xboole_0,dt_k3_xcmplx_0,dt_k4_card_1,dt_k4_euclid,dt_k4_finseq_1,dt_k4_finseq_2,dt_k4_finseqop,dt_k4_relset_1,dt_k5_euclid,dt_k5_finseq_1,dt_k5_numbers,dt_k5_ordinal2,dt_k6_domain_1,dt_k6_margrel1,dt_k7_domain_1,dt_k7_finseq_1,dt_k8_finseq_1,dt_k8_funct_2,dt_k9_finseq_1,dt_k9_relat_1,dt_m1_bintree2,dt_m1_finseq_1,dt_m1_finseq_2,dt_m1_relset_1,dt_m1_subset_1,dt_m1_trees_1,dt_m1_trees_4,dt_m2_finseq_1,dt_m2_finseq_2,dt_m2_relset_1,dt_m2_subset_1,dt_m2_trees_2,existence_m1_bintree2,existence_m1_finseq_1,existence_m1_finseq_2,existence_m1_relset_1,existence_m1_subset_1,existence_m1_trees_1,existence_m1_trees_4,existence_m2_finseq_1,existence_m2_finseq_2,existence_m2_relset_1,existence_m2_subset_1,existence_m2_trees_2,fc10_finset_1,fc10_xreal_0,fc11_finseq_1,fc11_finset_1,fc11_xreal_0,fc12_finseq_1,fc12_xreal_0,fc13_finseq_1,fc13_finset_1,fc14_finseq_1,fc14_finset_1,fc16_finseq_1,fc17_finseq_1,fc1_finseq_1,fc1_finset_1,fc1_margrel1,fc1_nat_1,fc1_ordinal2,fc1_subset_1,fc21_xreal_0,fc22_xreal_0,fc23_xreal_0,fc24_xreal_0,fc2_bintree2,fc2_finseq_1,fc2_finset_1,fc2_nat_1,fc2_subset_1,fc3_finseq_1,fc3_margrel1,fc3_nat_1,fc3_subset_1,fc3_xreal_0,fc4_finseq_1,fc4_nat_1,fc4_subset_1,fc4_xreal_0,fc5_trees_2,fc7_xreal_0,fc8_xreal_0,fc9_finseq_1,fc9_finset_1,fc9_xreal_0,fraenkel_a_2_0_trees_2,fraenkel_a_2_11_bintree2,fraenkel_a_2_12_bintree2,fraenkel_a_2_13_bintree2,fraenkel_a_2_14_bintree2,idempotence_k2_xboole_0,idempotence_k3_xboole_0,irreflexivity_r1_subset_1,rc1_bintree1,rc1_bintree2,rc1_finseq_1,rc1_finset_1,rc1_margrel1,rc1_nat_1,rc1_subset_1,rc1_trees_2,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_margrel1,rc2_nat_1,rc2_subset_1,rc2_xreal_0,rc3_finseq_1,rc3_finset_1,rc3_nat_1,rc3_trees_9,rc3_xreal_0,rc4_finseq_1,rc4_finset_1,rc4_xreal_0,rc5_trees_9,rc6_finseq_1,rc6_trees_9,rc7_finseq_1,rc8_finseq_1,rc9_trees_2,redefinition_k12_finseq_1,redefinition_k13_binarith,redefinition_k1_nat_1,redefinition_k2_finseq_1,redefinition_k2_funct_2,redefinition_k2_nat_1,redefinition_k3_finseq_1,redefinition_k3_finseq_2,redefinition_k3_series_1,redefinition_k4_card_1,redefinition_k4_finseq_1,redefinition_k4_finseqop,redefinition_k4_relset_1,redefinition_k5_euclid,redefinition_k5_numbers,redefinition_k6_domain_1,redefinition_k7_domain_1,redefinition_k8_finseq_1,redefinition_k8_funct_2,redefinition_k9_finseq_1,redefinition_m1_bintree2,redefinition_m1_trees_1,redefinition_m1_trees_4,redefinition_m2_finseq_1,redefinition_m2_finseq_2,redefinition_m2_relset_1,redefinition_m2_subset_1,redefinition_r1_subset_1,redefinition_r2_wellord2,reflexivity_r1_tarski,reflexivity_r1_xreal_0,reflexivity_r2_wellord2,s1_nat_1__e5_29__bintree2,s3_funct_2__e26_29_2__bintree2,s3_funct_2__e28_29_2__bintree2,spc0_boole,spc1_boole,spc1_numerals,spc2_boole,spc2_numerals,symmetry_r1_subset_1,symmetry_r1_xboole_0,symmetry_r2_wellord2,t10_bintree2,t13_finseq_2,t13_finset_1,t16_funct_2,t17_qc_lang4,t19_finseq_2,t1_arithm,t1_boole,t1_numerals,t1_real,t1_subset,t20_finseq_2,t22_finseq_2,t23_scmfsa_7,t29_power,t2_arithm,t2_boole,t2_euclid,t2_real,t2_subset,t2_tarski,t30_power,t32_power,t3_arithm,t3_real,t3_subset,t45_funct_2,t4_real,t4_subset,t53_card_2,t55_finseq_1,t59_finseq_1,t5_binari_3,t5_real,t5_subset,t60_card_1,t6_boole,t6_finseq_1,t6_real,t70_finseq_2,t79_card_1,t7_boole,t7_real,t81_card_1,t8_boole,t8_real,t8_xboole_1]), [file(bintree2,t18_bintree2)]). fof(antisymmetry_r2_hidden,axiom,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), []). fof(cc1_bintree1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_bintree1(A) ) => ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_trees_2(A) ) ) ), file(bintree1,cc1_bintree1), []). fof(cc1_bintree2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_bintree2(A) ) => ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_trees_2(A) & v2_trees_9(A) & v1_bintree1(A) ) ) ), file(bintree2,cc1_bintree2), []). 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_finset_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_finset_1(A) ) ), file(finset_1,cc1_finset_1), []). fof(cc1_margrel1,axiom,( ! [A] : ( m1_subset_1(A,k6_margrel1) => v2_margrel1(A) ) ), file(margrel1,cc1_margrel1), []). 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_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_trees_9,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_finset_1(A) & v1_trees_1(A) ) => ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_trees_2(A) ) ) ), file(trees_9,cc1_trees_9), []). fof(cc1_xreal_0,axiom,( ! [A] : ( v4_ordinal2(A) => v1_xreal_0(A) ) ), file(xreal_0,cc1_xreal_0), []). 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_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_xreal_0,axiom,( ! [A] : ( v1_xreal_0(A) => v1_xcmplx_0(A) ) ), file(xreal_0,cc2_xreal_0), []). 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_trees_9,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_trees_2(A) ) => ( ~ v1_xboole_0(A) & v1_trees_1(A) & v2_trees_9(A) ) ) ), file(trees_9,cc3_trees_9), []). 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_xreal_0,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v2_xreal_0(A) ) ) ), file(xreal_0,cc4_xreal_0), []). fof(cc5_xreal_0,axiom,( ! [A] : ( ( v1_xreal_0(A) & v3_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) ) ) ), file(xreal_0,cc5_xreal_0), []). fof(cc6_xreal_0,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v3_xreal_0(A) ) ) ), file(xreal_0,cc6_xreal_0), []). fof(cc7_xreal_0,axiom,( ! [A] : ( ( v1_xboole_0(A) & v1_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc7_xreal_0), []). fof(cc8_xreal_0,axiom,( ! [A] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(xreal_0,cc8_xreal_0), []). fof(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_k7_domain_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => k7_domain_1(A,B,C) = k7_domain_1(A,C,B) ) ), file(domain_1,k7_domain_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(d10_xboole_0,axiom,( ! [A,B] : ( A = B <=> ( r1_tarski(A,B) & r1_tarski(B,A) ) ) ), file(xboole_0,d10_xboole_0), []). fof(d11_finseq_1,axiom,( ! [A,B] : ( B = k13_finseq_1(A) <=> ! [C] : ( r2_hidden(C,B) <=> m2_finseq_1(C,A) ) ) ), file(finseq_1,d11_finseq_1), []). fof(d12_margrel1,axiom,( k6_margrel1 = k2_tarski(0,1) ), file(margrel1,d12_margrel1), []). fof(d1_funct_2,axiom,( ! [A,B,C] : ( m2_relset_1(C,A,B) => ( ( ( B = k1_xboole_0 => A = k1_xboole_0 ) => ( v1_funct_2(C,A,B) <=> A = k4_relset_1(A,B,C) ) ) & ( B = k1_xboole_0 => ( A = k1_xboole_0 | ( v1_funct_2(C,A,B) <=> C = k1_xboole_0 ) ) ) ) ) ), file(funct_2,d1_funct_2), []). 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_bintree2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => ( v1_bintree2(A) <=> A = k3_finseq_2(k7_domain_1(k5_numbers,0,1)) ) ) ), file(bintree2,d2_bintree2), []). fof(d2_tarski,axiom,( ! [A,B,C] : ( C = k2_tarski(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( D = A | D = B ) ) ) ), file(tarski,d2_tarski), []). fof(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_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( B = k3_finseq_1(A) <=> k2_finseq_1(B) = k1_relat_1(A) ) ) ) ), file(finseq_1,d3_finseq_1), []). fof(d3_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(d4_euclid,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k4_euclid(A) = k4_finseqop(k1_numbers,A,0) ) ), file(euclid,d4_euclid), []). fof(d4_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( m1_finseq_1(B,A) <=> r1_tarski(k2_relat_1(B),A) ) ) ), file(finseq_1,d4_finseq_1), []). fof(d6_trees_2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => k2_trees_2(A,B) = a_2_0_trees_2(A,B) ) ) ), file(trees_2,d6_trees_2), []). 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_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( v2_funct_1(A) <=> ! [B,C] : ( ( r2_hidden(B,k1_relat_1(A)) & r2_hidden(C,k1_relat_1(A)) & k1_funct_1(A,B) = k1_funct_1(A,C) ) => B = C ) ) ) ), file(funct_1,d8_funct_1), []). 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_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_finseq_1,axiom,( $true ), file(finseq_1,k13_finseq_1), []). fof(dt_k1_card_1,axiom,( ! [A] : v1_card_1(k1_card_1(A)) ), file(card_1,k1_card_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_relat_1,axiom,( $true ), file(relat_1,k1_relat_1), []). fof(dt_k1_tarski,axiom,( $true ), file(tarski,k1_tarski), []). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), []). fof(dt_k1_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), []). fof(dt_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_finseq_2,axiom,( ! [A,B] : ( v4_ordinal2(A) => ( v1_relat_1(k2_finseq_2(A,B)) & v1_funct_1(k2_finseq_2(A,B)) & v1_finseq_1(k2_finseq_2(A,B)) ) ) ), file(finseq_2,k2_finseq_2), []). fof(dt_k2_funct_2,axiom,( ! [A,B,C,D] : ( ( v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) ) => m1_subset_1(k2_funct_2(A,B,C,D),k1_zfmisc_1(B)) ) ), file(funct_2,k2_funct_2), []). 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_relat_1,axiom,( $true ), file(relat_1,k2_relat_1), []). fof(dt_k2_tarski,axiom,( $true ), file(tarski,k2_tarski), []). fof(dt_k2_trees_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) & m1_subset_1(B,k5_numbers) ) => m2_trees_2(k2_trees_2(A,B),A) ) ), file(trees_2,k2_trees_2), []). 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_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => m2_subset_1(k3_finseq_1(A),k1_numbers,k5_numbers) ) ), file(finseq_1,k3_finseq_1), []). fof(dt_k3_finseq_2,axiom,( ! [A] : ( ~ v1_xboole_0(k3_finseq_2(A)) & m1_finseq_2(k3_finseq_2(A),A) ) ), file(finseq_2,k3_finseq_2), []). fof(dt_k3_power,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => v1_xreal_0(k3_power(A,B)) ) ), file(power,k3_power), []). fof(dt_k3_series_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k3_series_1(A,B),k1_numbers,k5_numbers) ) ), file(series_1,k3_series_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_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => m2_finseq_1(k4_euclid(A),k1_numbers) ) ), file(euclid,k4_euclid), []). fof(dt_k4_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => m1_subset_1(k4_finseq_1(A),k1_zfmisc_1(k5_numbers)) ) ), file(finseq_1,k4_finseq_1), []). fof(dt_k4_finseq_2,axiom,( ! [A,B] : ( v4_ordinal2(A) => m1_finseq_2(k4_finseq_2(A,B),B) ) ), file(finseq_2,k4_finseq_2), []). fof(dt_k4_finseqop,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,k5_numbers) & m1_subset_1(C,A) ) => m2_finseq_2(k4_finseqop(A,B,C),A,k4_finseq_2(B,A)) ) ), file(finseqop,k4_finseqop), []). fof(dt_k4_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => m1_subset_1(k4_relset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(relset_1,k4_relset_1), []). fof(dt_k5_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => m2_finseq_2(k5_euclid(A),k1_numbers,k1_euclid(A)) ) ), file(euclid,k5_euclid), []). 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_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_margrel1,axiom,( $true ), file(margrel1,k6_margrel1), []). fof(dt_k7_domain_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => m1_subset_1(k7_domain_1(A,B,C),k1_zfmisc_1(A)) ) ), file(domain_1,k7_domain_1), []). fof(dt_k7_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( v1_relat_1(k7_finseq_1(A,B)) & v1_funct_1(k7_finseq_1(A,B)) & v1_finseq_1(k7_finseq_1(A,B)) ) ) ), file(finseq_1,k7_finseq_1), []). fof(dt_k8_finseq_1,axiom,( ! [A,B,C] : ( ( m1_finseq_1(B,A) & m1_finseq_1(C,A) ) => m2_finseq_1(k8_finseq_1(A,B,C),A) ) ), file(finseq_1,k8_finseq_1), []). fof(dt_k8_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_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_relat_1,axiom,( $true ), file(relat_1,k9_relat_1), []). fof(dt_m1_bintree2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_bintree1(A) ) => ! [B] : ( m1_bintree2(B,A) => m2_finseq_1(B,k6_margrel1) ) ) ), file(bintree2,m1_bintree2), []). 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_m1_trees_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => ! [B] : ( m1_trees_1(B,A) => m2_finseq_1(B,k5_numbers) ) ) ), file(trees_1,m1_trees_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(dt_m2_trees_2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => ! [B] : ( m2_trees_2(B,A) => m1_subset_1(B,k1_zfmisc_1(A)) ) ) ), file(trees_2,m2_trees_2), []). fof(existence_m1_bintree2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_bintree1(A) ) => ? [B] : m1_bintree2(B,A) ) ), file(bintree2,m1_bintree2), []). 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_m1_trees_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => ? [B] : m1_trees_1(B,A) ) ), file(trees_1,m1_trees_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(existence_m2_trees_2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => ? [B] : m2_trees_2(B,A) ) ), file(trees_2,m2_trees_2), []). 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_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_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_finset_1(A) ) => v1_finset_1(k2_relat_1(A)) ) ), file(finseq_1,fc11_finseq_1), []). 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_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_finseq_1,axiom,( ! [A] : ( ~ v1_xboole_0(k5_finseq_1(A)) & v1_relat_1(k5_finseq_1(A)) & v1_funct_1(k5_finseq_1(A)) & v1_finset_1(k5_finseq_1(A)) & v1_finseq_1(k5_finseq_1(A)) ) ), file(finseq_1,fc12_finseq_1), []). 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_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & ~ v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( ~ v1_xboole_0(k7_finseq_1(A,B)) & v1_relat_1(k7_finseq_1(A,B)) & v1_funct_1(k7_finseq_1(A,B)) & v1_finset_1(k7_finseq_1(A,B)) & v1_finseq_1(k7_finseq_1(A,B)) ) ) ), file(finseq_1,fc13_finseq_1), []). fof(fc13_finset_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(B) ) => v1_finset_1(k9_relat_1(A,B)) ) ), file(finset_1,fc13_finset_1), []). fof(fc14_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & ~ v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( ~ v1_xboole_0(k7_finseq_1(B,A)) & v1_relat_1(k7_finseq_1(B,A)) & v1_funct_1(k7_finseq_1(B,A)) & v1_finset_1(k7_finseq_1(B,A)) & v1_finseq_1(k7_finseq_1(B,A)) ) ) ), file(finseq_1,fc14_finseq_1), []). 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(fc16_finseq_1,axiom,( ! [A] : ( ~ v1_xboole_0(k13_finseq_1(A)) & v1_fraenkel(k13_finseq_1(A)) ) ), file(finseq_1,fc16_finseq_1), []). fof(fc17_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) => v1_finset_1(k1_relat_1(A)) ) ), file(finseq_1,fc17_finseq_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_margrel1,axiom, ( v1_xboole_0(k1_xboole_0) & v1_margrel1(k1_xboole_0) ), file(margrel1,fc1_margrel1), []). 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_ordinal2,axiom, ( v1_ordinal1(k5_ordinal2) & v2_ordinal1(k5_ordinal2) & v3_ordinal1(k5_ordinal2) & ~ v1_xboole_0(k5_ordinal2) ), file(ordinal2,fc1_ordinal2), []). fof(fc1_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ), file(subset_1,fc1_subset_1), []). fof(fc21_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & ~ v2_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc21_xreal_0), []). fof(fc22_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(B,A)) & v1_xreal_0(k3_xcmplx_0(B,A)) & ~ v2_xreal_0(k3_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc22_xreal_0), []). fof(fc23_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & ~ v3_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc23_xreal_0), []). fof(fc24_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & ~ v3_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc24_xreal_0), []). fof(fc2_bintree2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_bintree1(A) & m1_subset_1(B,k5_numbers) ) => v1_finset_1(k2_trees_2(A,B)) ) ), file(bintree2,fc2_bintree2), []). 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_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(fc3_finseq_1,axiom,( ! [A] : ( v1_relat_1(k5_finseq_1(A)) & v1_funct_1(k5_finseq_1(A)) ) ), file(finseq_1,fc3_finseq_1), []). fof(fc3_margrel1,axiom,( ~ v1_xboole_0(k6_margrel1) ), file(margrel1,fc3_margrel1), []). 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_subset_1,axiom,( ! [A,B] : ~ v1_xboole_0(k2_tarski(A,B)) ), file(subset_1,fc3_subset_1), []). 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_finseq_1,axiom,( ! [A] : ( v1_relat_1(k5_finseq_1(A)) & v1_funct_1(k5_finseq_1(A)) & v1_finset_1(k5_finseq_1(A)) & v1_finseq_1(k5_finseq_1(A)) ) ), file(finseq_1,fc4_finseq_1), []). 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(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_trees_2,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_finset_1(A) ) => v1_finset_1(k1_relat_1(A)) ) ), file(trees_2,fc5_trees_2), []). 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_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_finseq_1,axiom,( ! [A] : ~ v1_xboole_0(k13_finseq_1(A)) ), file(finseq_1,fc9_finseq_1), []). 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_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_2_0_trees_2,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(B) & v1_trees_1(B) & m2_subset_1(C,k1_numbers,k5_numbers) ) => ( r2_hidden(A,a_2_0_trees_2(B,C)) <=> ? [D] : ( m1_trees_1(D,B) & A = D & k3_finseq_1(D) = C ) ) ) ), file(trees_2,a_2_0_trees_2), []). fof(fraenkel_a_2_11_bintree2,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(B) & v1_trees_1(B) & v1_bintree2(B) & m2_subset_1(C,k1_numbers,k5_numbers) ) => ( r2_hidden(A,a_2_11_bintree2(B,C)) <=> ? [D] : ( m1_bintree2(D,B) & A = D & k3_finseq_1(D) = k1_nat_1(C,1) & k1_funct_1(D,k1_nat_1(C,1)) = 0 ) ) ) ), file(bintree2,a_2_11_bintree2), []). fof(fraenkel_a_2_12_bintree2,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(B) & v1_trees_1(B) & v1_bintree2(B) & m2_subset_1(C,k1_numbers,k5_numbers) ) => ( r2_hidden(A,a_2_12_bintree2(B,C)) <=> ? [D] : ( m1_bintree2(D,B) & A = D & k3_finseq_1(D) = k1_nat_1(C,1) & k1_funct_1(D,k1_nat_1(C,1)) = 1 ) ) ) ), file(bintree2,a_2_12_bintree2), []). fof(fraenkel_a_2_13_bintree2,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(B) & v1_trees_1(B) & v1_bintree2(B) & m2_subset_1(C,k1_numbers,k5_numbers) ) => ( r2_hidden(A,a_2_13_bintree2(B,C)) <=> ? [D] : ( m1_bintree2(D,B) & A = D & k3_finseq_1(D) = k1_nat_1(C,1) ) ) ) ), file(bintree2,a_2_13_bintree2), []). fof(fraenkel_a_2_14_bintree2,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(B) & v1_trees_1(B) & v1_bintree2(B) & m2_subset_1(C,k1_numbers,k5_numbers) ) => ( r2_hidden(A,a_2_14_bintree2(B,C)) <=> ? [D] : ( m1_bintree2(D,B) & A = D & k3_finseq_1(D) = C ) ) ) ), file(bintree2,a_2_14_bintree2), []). 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(irreflexivity_r1_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ~ r1_subset_1(A,A) ) ), file(subset_1,r1_subset_1), []). fof(rc1_bintree1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) & v1_trees_1(A) & v1_bintree1(A) ) ), file(bintree1,rc1_bintree1), []). fof(rc1_bintree2,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_trees_2(A) & v2_trees_9(A) & v1_bintree1(A) & v1_bintree2(A) ) ), file(bintree2,rc1_bintree2), []). 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_finset_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) ) ), file(finset_1,rc1_finset_1), []). fof(rc1_margrel1,axiom,( ? [A] : v1_margrel1(A) ), file(margrel1,rc1_margrel1), []). 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_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(rc1_trees_2,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_trees_2(A) ) ), file(trees_2,rc1_trees_2), []). fof(rc1_xreal_0,axiom,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(xreal_0,rc1_xreal_0), []). 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_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_margrel1,axiom,( ? [A] : v2_margrel1(A) ), file(margrel1,rc2_margrel1), []). 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_subset_1,axiom,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) ) ), file(subset_1,rc2_subset_1), []). 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_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(rc3_trees_9,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) & v1_trees_1(A) & v1_trees_2(A) & v2_trees_9(A) ) ), file(trees_9,rc3_trees_9), []). 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_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(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_trees_9,axiom,( ! [A] : ? [B] : ( m1_finseq_1(B,A) & v1_relat_1(B) & v1_funct_1(B) & v2_funct_1(B) & v1_xboole_0(B) & v1_finset_1(B) & v1_finseq_1(B) ) ), file(trees_9,rc5_trees_9), []). 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(rc6_trees_9,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => ? [B] : ( m1_subset_1(B,A) & v1_relat_1(B) & v1_funct_1(B) & v2_funct_1(B) & v1_xboole_0(B) & v1_finset_1(B) & v1_finseq_1(B) ) ) ), file(trees_9,rc6_trees_9), []). 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(rc9_trees_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ? [C] : ( m1_relset_1(C,A,B) & v1_relat_1(C) & ~ v1_xboole_0(C) ) ) ), file(trees_2,rc9_trees_2), []). fof(redefinition_k12_finseq_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => k12_finseq_1(A,B) = k5_finseq_1(B) ) ), file(finseq_1,k12_finseq_1), []). fof(redefinition_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_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_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => k2_finseq_1(A) = k1_finseq_1(A) ) ), file(finseq_1,k2_finseq_1), []). fof(redefinition_k2_funct_2,axiom,( ! [A,B,C,D] : ( ( v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) ) => k2_funct_2(A,B,C,D) = k9_relat_1(C,D) ) ), file(funct_2,k2_funct_2), []). 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_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => k3_finseq_1(A) = k1_card_1(A) ) ), file(finseq_1,k3_finseq_1), []). fof(redefinition_k3_finseq_2,axiom,( ! [A] : k3_finseq_2(A) = k13_finseq_1(A) ), file(finseq_2,k3_finseq_2), []). fof(redefinition_k3_series_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k3_series_1(A,B) = k3_power(A,B) ) ), file(series_1,k3_series_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_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => k4_finseq_1(A) = k1_relat_1(A) ) ), file(finseq_1,k4_finseq_1), []). fof(redefinition_k4_finseqop,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,k5_numbers) & m1_subset_1(C,A) ) => k4_finseqop(A,B,C) = k2_finseq_2(B,C) ) ), file(finseqop,k4_finseqop), []). fof(redefinition_k4_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => k4_relset_1(A,B,C) = k1_relat_1(C) ) ), file(relset_1,k4_relset_1), []). fof(redefinition_k5_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => k5_euclid(A) = k4_euclid(A) ) ), file(euclid,k5_euclid), []). fof(redefinition_k5_numbers,axiom,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), []). 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_k7_domain_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => k7_domain_1(A,B,C) = k2_tarski(B,C) ) ), file(domain_1,k7_domain_1), []). fof(redefinition_k8_finseq_1,axiom,( ! [A,B,C] : ( ( m1_finseq_1(B,A) & m1_finseq_1(C,A) ) => k8_finseq_1(A,B,C) = k7_finseq_1(B,C) ) ), file(finseq_1,k8_finseq_1), []). fof(redefinition_k8_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_k9_finseq_1,axiom,( ! [A] : k9_finseq_1(A) = k5_finseq_1(A) ), file(finseq_1,k9_finseq_1), []). fof(redefinition_m1_bintree2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_bintree1(A) ) => ! [B] : ( m1_bintree2(B,A) <=> m1_subset_1(B,A) ) ) ), file(bintree2,m1_bintree2), []). fof(redefinition_m1_trees_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => ! [B] : ( m1_trees_1(B,A) <=> m1_subset_1(B,A) ) ) ), file(trees_1,m1_trees_1), []). 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_r1_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ( r1_subset_1(A,B) <=> r1_xboole_0(A,B) ) ) ), file(subset_1,r1_subset_1), []). fof(redefinition_r2_wellord2,axiom,( ! [A,B] : ( r2_wellord2(A,B) <=> r2_tarski(A,B) ) ), file(wellord2,r2_wellord2), []). 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_r2_wellord2,axiom,( ! [A,B] : r2_wellord2(A,A) ), file(wellord2,r2_wellord2), []). fof(s1_nat_1__e5_29__bintree2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) & v1_bintree2(A) ) => ( ( k1_card_1(k2_trees_2(A,0)) = k3_series_1(2,0) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( k1_card_1(k2_trees_2(A,B)) = k3_series_1(2,B) => k1_card_1(k2_trees_2(A,k1_nat_1(B,1))) = k3_series_1(2,k1_nat_1(B,1)) ) ) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => k1_card_1(k2_trees_2(A,B)) = k3_series_1(2,B) ) ) ) ), file(bintree2,s1_nat_1__e5_29__bintree2), []). fof(s3_funct_2__e26_29_2__bintree2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_finset_1(A) & ~ v1_xboole_0(B) & v1_finset_1(B) ) => ( ! [C] : ( m1_subset_1(C,B) => ? [D] : ( m1_subset_1(D,A) & ? [E] : ( v1_relat_1(E) & v1_funct_1(E) & v1_finseq_1(E) & E = C & D = k7_finseq_1(E,k13_binarith(k5_numbers,0)) ) ) ) => ? [C] : ( v1_funct_1(C) & v1_funct_2(C,B,A) & m2_relset_1(C,B,A) & ! [D] : ( m1_subset_1(D,B) => ? [F] : ( v1_relat_1(F) & v1_funct_1(F) & v1_finseq_1(F) & F = D & k8_funct_2(B,A,C,D) = k7_finseq_1(F,k13_binarith(k5_numbers,0)) ) ) ) ) ) ), file(bintree2,s3_funct_2__e26_29_2__bintree2), []). fof(s3_funct_2__e28_29_2__bintree2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_finset_1(A) & ~ v1_xboole_0(B) & v1_finset_1(B) ) => ( ! [C] : ( m1_subset_1(C,B) => ? [D] : ( m1_subset_1(D,A) & ? [E] : ( v1_relat_1(E) & v1_funct_1(E) & v1_finseq_1(E) & E = C & D = k7_finseq_1(E,k13_binarith(k5_numbers,1)) ) ) ) => ? [C] : ( v1_funct_1(C) & v1_funct_2(C,B,A) & m2_relset_1(C,B,A) & ! [D] : ( m1_subset_1(D,B) => ? [F] : ( v1_relat_1(F) & v1_funct_1(F) & v1_finseq_1(F) & F = D & k8_funct_2(B,A,C,D) = k7_finseq_1(F,k13_binarith(k5_numbers,1)) ) ) ) ) ) ), file(bintree2,s3_funct_2__e28_29_2__bintree2), []). 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_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ( r1_subset_1(A,B) => r1_subset_1(B,A) ) ) ), file(subset_1,r1_subset_1), []). 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_wellord2,axiom,( ! [A,B] : ( r2_wellord2(A,B) => r2_wellord2(B,A) ) ), file(wellord2,r2_wellord2), []). fof(t10_bintree2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => ( A = k3_finseq_2(k7_domain_1(k5_numbers,0,1)) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => r2_hidden(k5_euclid(B),k2_trees_2(A,B)) ) ) ) ), file(bintree2,t10_bintree2), []). fof(t13_finseq_2,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B,C] : ( m2_finseq_1(C,B) => ( r2_hidden(A,k4_finseq_1(C)) => r2_hidden(k1_funct_1(C,A),B) ) ) ) ), file(finseq_2,t13_finseq_2), []). fof(t13_finset_1,axiom,( ! [A,B] : ( ( r1_tarski(A,B) & v1_finset_1(B) ) => v1_finset_1(A) ) ), file(finset_1,t13_finset_1), []). fof(t16_funct_2,axiom,( ! [A,B,C] : ( ( v1_funct_1(C) & v1_funct_2(C,A,B) & m2_relset_1(C,A,B) ) => ( ! [D] : ~ ( r2_hidden(D,B) & ! [E] : ~ ( r2_hidden(E,A) & D = k1_funct_1(C,E) ) ) => ( B = k1_xboole_0 | k2_relat_1(C) = B ) ) ) ), file(funct_2,t16_funct_2), []). fof(t17_qc_lang4,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_trees_1(A) ) => k2_trees_2(A,0) = k1_tarski(k1_xboole_0) ) ), file(qc_lang4,t17_qc_lang4), []). fof(t19_finseq_2,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => k3_finseq_1(k7_finseq_1(B,k9_finseq_1(A))) = k1_nat_1(k3_finseq_1(B),1) ) ), file(finseq_2,t19_finseq_2), []). 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(t20_finseq_2,axiom,( ! [A,B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v1_finseq_1(C) ) => ! [D] : ( ( v1_relat_1(D) & v1_funct_1(D) & v1_finseq_1(D) ) => ( k7_finseq_1(C,k9_finseq_1(A)) = k7_finseq_1(D,k9_finseq_1(B)) => ( C = D & A = B ) ) ) ) ), file(finseq_2,t20_finseq_2), []). fof(t22_finseq_2,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m2_finseq_1(B,A) => ~ ( k3_finseq_1(B) != 0 & ! [C] : ( m2_finseq_1(C,A) => ! [D] : ( m1_subset_1(D,A) => B != k8_finseq_1(A,C,k12_finseq_1(A,D)) ) ) ) ) ) ), file(finseq_2,t22_finseq_2), []). fof(t23_scmfsa_7,axiom,( ! [A,B] : ( m2_finseq_1(B,A) => ! [C] : ( m2_finseq_1(C,A) => m2_finseq_1(k8_finseq_1(A,B,C),A) ) ) ), file(scmfsa_7,t23_scmfsa_7), []). fof(t29_power,axiom,( ! [A] : ( v1_xreal_0(A) => k3_power(A,0) = 1 ) ), file(power,t29_power), []). 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_euclid,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_finseq_2(B,k1_numbers,k1_euclid(A)) => k3_finseq_1(B) = A ) ) ), file(euclid,t2_euclid), []). 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(t30_power,axiom,( ! [A] : ( v1_xreal_0(A) => k3_power(A,1) = A ) ), file(power,t30_power), []). fof(t32_power,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ~ r1_xreal_0(A,0) => k3_power(A,k2_xcmplx_0(B,C)) = k3_xcmplx_0(k3_power(A,B),k3_power(A,C)) ) ) ) ) ), file(power,t32_power), []). fof(t3_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(1,A) = A ) ), file(arithm,t3_arithm), []). 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(t45_funct_2,axiom,( ! [A,B,C] : ( ( v1_funct_1(C) & v1_funct_2(C,A,B) & m2_relset_1(C,A,B) ) => ( ( B = k1_xboole_0 => A = k1_xboole_0 ) => k2_funct_2(A,B,C,A) = k2_relat_1(C) ) ) ), file(funct_2,t45_funct_2), []). 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(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(t55_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( B = k9_finseq_1(A) <=> ( k4_finseq_1(B) = k2_finseq_1(1) & k2_relat_1(B) = k1_tarski(A) ) ) ) ), file(finseq_1,t55_finseq_1), []). fof(t59_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => k1_funct_1(k7_finseq_1(B,k9_finseq_1(A)),k1_nat_1(k3_finseq_1(B),1)) = A ) ), file(finseq_1,t59_finseq_1), []). fof(t5_binari_3,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => r2_hidden(k5_euclid(A),k13_finseq_1(k6_margrel1)) ) ), file(binari_3,t5_binari_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(t60_card_1,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( ( r1_tarski(A,k1_relat_1(B)) & v2_funct_1(B) ) => r2_wellord2(A,k9_relat_1(B,A)) ) ) ), file(card_1,t60_card_1), []). fof(t6_boole,axiom,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), []). fof(t6_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => r2_hidden(k2_xcmplx_0(A,1),k2_finseq_1(k2_xcmplx_0(A,1))) ) ), file(finseq_1,t6_finseq_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(t70_finseq_2,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B,C] : ( r2_hidden(B,k2_finseq_1(A)) => k1_funct_1(k2_finseq_2(A,C),B) = C ) ) ), file(finseq_2,t70_finseq_2), []). 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(t81_card_1,axiom,( ! [A] : ( v1_finset_1(A) => ! [B] : ( v1_finset_1(B) => ( r2_wellord2(A,B) => k4_card_1(A) = k4_card_1(B) ) ) ) ), file(card_1,t81_card_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), []).