% Mizar problem: t8_scpinvar,scpinvar,348,13 fof(t8_scpinvar,conjecture,( ! [A] : ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) => ! [B] : ( ( v1_ami_3(B,k1_tarski(k4_numbers),k1_scmpds_2) & v1_scmpds_3(B) & v3_scmpds_4(B) & v3_scmpds_5(B) & m1_ami_1(B,k1_tarski(k4_numbers),k1_scmpds_2) ) => ! [C] : ( m1_scmpds_2(C) => ! [D] : ( m1_scmpds_2(D) => ! [E] : ( m1_scmpds_2(E) => ! [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) => ! [G] : ( m2_subset_1(G,k1_numbers,k5_numbers) => ! [H] : ( m2_subset_1(H,k1_numbers,k5_numbers) => ! [I] : ( m1_trees_4(I,k1_numbers,k6_wsierp_1) => ( ( r1_scpisort(I,A,H) & k3_finseq_1(I) = F & k2_scmpds_2(A,D) = 0 & k2_scmpds_2(A,C) = 0 & k2_scmpds_2(A,k1_scmp_gcd(G)) = k4_xcmplx_0(F) & k2_scmpds_2(A,E) = k1_nat_1(H,1) & ! [J] : ( m1_subset_1(J,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) => ( ( k2_scmpds_2(J,C) = 0 & ! [K] : ( m2_subset_1(K,k1_numbers,k5_numbers) => ( ~ r1_xreal_0(K,H) => k2_scmpds_2(J,k1_scmp_gcd(K)) = k2_scmpds_2(A,k1_scmp_gcd(K)) ) ) ) => ( ! [K] : ( m1_trees_4(K,k1_numbers,k6_wsierp_1) => ~ ( r1_scpisort(K,A,H) & k3_finseq_1(K) = k2_xcmplx_0(k2_scmpds_2(J,k1_scmp_gcd(G)),F) & k2_scmpds_2(J,D) = k7_wsierp_1(K) & k2_scmpds_2(J,E) = k1_nat_1(k1_nat_1(H,1),k3_finseq_1(K)) ) ) | r1_xreal_0(0,k2_scmpds_2(J,k1_scmp_gcd(G))) | ( k2_scmpds_2(k9_scmpds_4(B,J),C) = 0 & r1_scmpds_6(B,J) & r2_scmpds_6(B,J) & k2_scmpds_2(k9_scmpds_4(B,J),k1_scmp_gcd(G)) = k2_xcmplx_0(k2_scmpds_2(J,k1_scmp_gcd(G)),1) & ? [K] : ( m1_trees_4(K,k1_numbers,k6_wsierp_1) & r1_scpisort(K,A,H) & k3_finseq_1(K) = k2_xcmplx_0(k2_xcmplx_0(k2_scmpds_2(J,k1_scmp_gcd(G)),F),1) & k2_scmpds_2(k9_scmpds_4(B,J),E) = k1_nat_1(k1_nat_1(H,1),k3_finseq_1(K)) & k2_scmpds_2(k9_scmpds_4(B,J),D) = k7_wsierp_1(K) ) & ! [K] : ( m2_subset_1(K,k1_numbers,k5_numbers) => ( ~ r1_xreal_0(K,H) => k2_scmpds_2(k9_scmpds_4(B,J),k1_scmp_gcd(K)) = k2_scmpds_2(A,k1_scmp_gcd(K)) ) ) ) ) ) ) ) => ( r1_xreal_0(k4_card_1(B),0) | ( k2_scmpds_2(k9_scmpds_4(k2_scmpds_8(C,G,B),A),D) = k7_wsierp_1(I) & r1_scmpds_6(k2_scmpds_8(C,G,B),A) & r2_scmpds_6(k2_scmpds_8(C,G,B),A) ) ) ) ) ) ) ) ) ) ) ) ) ), inference(mizar_bg_added,[status(thm)],[abstractness_v1_ami_1,antisymmetry_r2_hidden,cc1_fraenkel,cc1_funct_1,cc1_int_1,cc1_relset_1,cc1_scmfsa_4,cc1_setfam_1,cc2_funct_1,cc2_int_1,cc3_int_1,cc4_int_1,commutativity_k1_nat_1,commutativity_k2_tarski,commutativity_k2_xboole_0,commutativity_k2_xcmplx_0,connectedness_r1_xreal_0,d1_scpisort,dt_g1_ami_1,dt_k13_finseq_1,dt_k15_rvsum_1,dt_k1_card_1,dt_k1_finseq_1,dt_k1_funct_1,dt_k1_funct_2,dt_k1_nat_1,dt_k1_numbers,dt_k1_scmp_gcd,dt_k1_scmpds_2,dt_k1_scmpds_8,dt_k1_tarski,dt_k1_wsierp_1,dt_k1_zfmisc_1,dt_k2_finseq_1,dt_k2_scmpds_2,dt_k2_scmpds_8,dt_k2_tarski,dt_k2_xboole_0,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k3_finseq_1,dt_k3_scmpds_2,dt_k3_tarski,dt_k4_card_1,dt_k4_card_3,dt_k4_numbers,dt_k4_xcmplx_0,dt_k5_numbers,dt_k5_ordinal2,dt_k6_finseq_1,dt_k6_wsierp_1,dt_k6_xcmplx_0,dt_k7_ami_1,dt_k7_wsierp_1,dt_k8_funct_2,dt_k9_scmpds_4,dt_l1_ami_1,dt_l1_struct_0,dt_m1_ami_1,dt_m1_finseq_1,dt_m1_relset_1,dt_m1_scmpds_2,dt_m1_subset_1,dt_m1_trees_4,dt_m2_finseq_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_ami_1,dt_u1_struct_0,dt_u2_ami_1,dt_u3_ami_1,dt_u4_ami_1,dt_u5_ami_1,dt_u6_ami_1,existence_l1_ami_1,existence_l1_struct_0,existence_m1_ami_1,existence_m1_finseq_1,existence_m1_relset_1,existence_m1_scmpds_2,existence_m1_subset_1,existence_m1_trees_4,existence_m2_finseq_1,existence_m2_relset_1,existence_m2_subset_1,fc10_ami_1,fc1_fraenkel,fc1_int_1,fc1_numbers,fc1_ordinal2,fc1_scmpds_2,fc1_scmpds_8,fc2_ami_1,fc2_fraenkel,fc2_scmpds_2,fc2_scmpds_8,fc2_setfam_1,fc2_wsierp_1,fc3_ami_1,fc3_int_1,fc3_setfam_1,fc4_ami_1,fc4_int_1,fc4_numbers,fc4_scmpds_2,fc4_setfam_1,fc5_int_1,fc5_scmpds_2,fc6_int_1,fc8_int_1,fc9_int_1,free_g1_ami_1,idempotence_k2_xboole_0,involutiveness_k4_xcmplx_0,rc1_ami_1,rc1_fraenkel,rc1_funct_1,rc1_int_1,rc1_scmfsa_4,rc1_setfam_1,rc2_ami_1,rc2_funct_1,rc2_int_1,rc3_ami_1,rc3_funct_1,rc5_ami_1,rc6_ami_1,rc7_ami_1,rc8_ami_1,rc9_ami_1,redefinition_k1_nat_1,redefinition_k1_wsierp_1,redefinition_k2_finseq_1,redefinition_k2_scmpds_2,redefinition_k3_finseq_1,redefinition_k4_card_1,redefinition_k5_numbers,redefinition_k6_wsierp_1,redefinition_k7_wsierp_1,redefinition_k8_funct_2,redefinition_m1_trees_4,redefinition_m2_finseq_1,redefinition_m2_relset_1,redefinition_m2_subset_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,s1_scmpds_8__e19_10__scpinvar,s1_scpinvar__e9_10__scpinvar,spc1_numerals,t102_rvsum_1,t10_finseq_2,t148_xreal_1,t19_nat_1,t1_arithm,t1_numerals,t1_real,t1_subset,t1_xreal_1,t20_int_1,t2_real,t2_subset,t32_finseq_1,t3_finseq_1,t3_real,t3_scpisort,t3_subset,t4_arithm,t4_real,t4_scmpds_8,t4_subset,t5_real,t5_scmp_gcd,t5_subset,t6_real,t7_real,t7_scpinvar,t8_real]), [file(scpinvar,t8_scpinvar)]). fof(abstractness_v1_ami_1,axiom,( ! [A,B] : ( l1_ami_1(B,A) => ( v1_ami_1(B,A) => B = g1_ami_1(A,u1_struct_0(B),u1_ami_1(A,B),u2_ami_1(A,B),u3_ami_1(A,B),u4_ami_1(A,B),u5_ami_1(A,B),u6_ami_1(A,B)) ) ) ), file(ami_1,v1_ami_1), []). fof(antisymmetry_r2_hidden,axiom,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), []). fof(cc1_fraenkel,axiom,( ! [A] : ( v1_fraenkel(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_relat_1(B) & v1_funct_1(B) ) ) ) ), file(fraenkel,cc1_fraenkel), []). fof(cc1_funct_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_funct_1(A) ) ), file(funct_1,cc1_funct_1), []). fof(cc1_int_1,axiom,( ! [A] : ( m1_subset_1(A,k4_numbers) => v1_int_1(A) ) ), file(int_1,cc1_int_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_scmfsa_4,axiom,( ! [A,B] : ( l1_ami_1(B,A) => ! [C] : ( m1_ami_1(C,A,B) => ( v1_relat_1(C) & v1_funct_1(C) & v1_finset_1(C) ) ) ) ), file(scmfsa_4,cc1_scmfsa_4), []). fof(cc1_setfam_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_setfam_1(A) ) => ! [B] : ( m1_subset_1(B,A) => ~ v1_xboole_0(B) ) ) ), file(setfam_1,cc1_setfam_1), []). fof(cc2_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ) ), file(funct_1,cc2_funct_1), []). fof(cc2_int_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_int_1(A) ) ) ), file(int_1,cc2_int_1), []). fof(cc3_int_1,axiom,( ! [A] : ( v4_ordinal2(A) => v1_int_1(A) ) ), file(int_1,cc3_int_1), []). fof(cc4_int_1,axiom,( ! [A] : ( v1_int_1(A) => ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(int_1,cc4_int_1), []). fof(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_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(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(d1_scpisort,axiom,( ! [A] : ( m2_finseq_1(A,k4_numbers) => ! [B] : ( m1_subset_1(B,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( r1_scpisort(A,B,C) <=> ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,D) & r1_xreal_0(D,k3_finseq_1(A)) ) => k1_funct_1(A,D) = k2_scmpds_2(B,k1_scmp_gcd(k1_nat_1(C,D))) ) ) ) ) ) ) ), file(scpisort,d1_scpisort), []). fof(dt_g1_ami_1,axiom,( ! [A,B,C,D,E,F,G,H] : ( ( m1_subset_1(C,B) & m1_subset_1(D,k1_zfmisc_1(B)) & ~ v1_xboole_0(E) & ~ v1_xboole_0(F) & m1_subset_1(F,k1_zfmisc_1(k2_zfmisc_1(E,k13_finseq_1(k2_xboole_0(k3_tarski(A),B))))) & v1_funct_1(G) & v1_funct_2(G,B,k2_xboole_0(A,k2_tarski(F,D))) & m1_relset_1(G,B,k2_xboole_0(A,k2_tarski(F,D))) & v1_funct_1(H) & v1_funct_2(H,F,k1_funct_2(k4_card_3(G),k4_card_3(G))) & m1_relset_1(H,F,k1_funct_2(k4_card_3(G),k4_card_3(G))) ) => ( v1_ami_1(g1_ami_1(A,B,C,D,E,F,G,H),A) & l1_ami_1(g1_ami_1(A,B,C,D,E,F,G,H),A) ) ) ), file(ami_1,g1_ami_1), []). fof(dt_k13_finseq_1,axiom,( $true ), file(finseq_1,k13_finseq_1), []). fof(dt_k15_rvsum_1,axiom,( ! [A] : ( m1_finseq_1(A,k1_numbers) => m1_subset_1(k15_rvsum_1(A),k1_numbers) ) ), file(rvsum_1,k15_rvsum_1), []). fof(dt_k1_card_1,axiom,( ! [A] : v1_card_1(k1_card_1(A)) ), file(card_1,k1_card_1), []). fof(dt_k1_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_funct_2,axiom,( $true ), file(funct_2,k1_funct_2), []). 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_scmp_gcd,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => m1_scmpds_2(k1_scmp_gcd(A)) ) ), file(scmp_gcd,k1_scmp_gcd), []). fof(dt_k1_scmpds_2,axiom, ( v1_ami_1(k1_scmpds_2,k1_tarski(k4_numbers)) & l1_ami_1(k1_scmpds_2,k1_tarski(k4_numbers)) ), file(scmpds_2,k1_scmpds_2), []). fof(dt_k1_scmpds_8,axiom,( ! [A] : ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) => m1_subset_1(k1_scmpds_8(A),k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) ) ), file(scmpds_8,k1_scmpds_8), []). fof(dt_k1_tarski,axiom,( $true ), file(tarski,k1_tarski), []). fof(dt_k1_wsierp_1,axiom,( ! [A,B] : ( ( m1_finseq_1(A,k1_numbers) & m1_subset_1(B,k5_numbers) ) => m1_subset_1(k1_wsierp_1(A,B),k1_numbers) ) ), file(wsierp_1,k1_wsierp_1), []). 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_scmpds_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) & m1_scmpds_2(B) ) => v1_int_1(k2_scmpds_2(A,B)) ) ), file(scmpds_2,k2_scmpds_2), []). fof(dt_k2_scmpds_8,axiom,( ! [A,B,C] : ( ( m1_scmpds_2(A) & v1_int_1(B) & v1_ami_3(C,k1_tarski(k4_numbers),k1_scmpds_2) & v1_scmpds_3(C) & m1_ami_1(C,k1_tarski(k4_numbers),k1_scmpds_2) ) => ( v1_ami_3(k2_scmpds_8(A,B,C),k1_tarski(k4_numbers),k1_scmpds_2) & v1_scmpds_3(k2_scmpds_8(A,B,C)) & m1_ami_1(k2_scmpds_8(A,B,C),k1_tarski(k4_numbers),k1_scmpds_2) ) ) ), file(scmpds_8,k2_scmpds_8), []). 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_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_scmpds_2,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => m1_scmpds_2(k3_scmpds_2(A,B)) ) ), file(scmpds_2,k3_scmpds_2), []). fof(dt_k3_tarski,axiom,( $true ), file(tarski,k3_tarski), []). 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_card_3,axiom,( $true ), file(card_3,k4_card_3), []). fof(dt_k4_numbers,axiom,( $true ), file(numbers,k4_numbers), []). fof(dt_k4_xcmplx_0,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_xcmplx_0(k4_xcmplx_0(A)) ) ), file(xcmplx_0,k4_xcmplx_0), []). 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_finseq_1,axiom,( ! [A] : ( v1_xboole_0(k6_finseq_1(A)) & m2_finseq_1(k6_finseq_1(A),A) ) ), file(finseq_1,k6_finseq_1), []). fof(dt_k6_wsierp_1,axiom, ( ~ v1_xboole_0(k6_wsierp_1) & m1_subset_1(k6_wsierp_1,k1_zfmisc_1(k1_numbers)) ), file(wsierp_1,k6_wsierp_1), []). fof(dt_k6_xcmplx_0,axiom,( $true ), file(xcmplx_0,k6_xcmplx_0), []). fof(dt_k7_ami_1,axiom,( $true ), file(ami_1,k7_ami_1), []). fof(dt_k7_wsierp_1,axiom,( ! [A] : ( m1_finseq_1(A,k6_wsierp_1) => m2_subset_1(k7_wsierp_1(A),k1_numbers,k6_wsierp_1) ) ), file(wsierp_1,k7_wsierp_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_scmpds_4,axiom,( ! [A,B] : ( ( v1_ami_3(A,k1_tarski(k4_numbers),k1_scmpds_2) & v1_scmpds_3(A) & m1_ami_1(A,k1_tarski(k4_numbers),k1_scmpds_2) & m1_subset_1(B,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) ) => m1_subset_1(k9_scmpds_4(A,B),k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) ) ), file(scmpds_4,k9_scmpds_4), []). fof(dt_l1_ami_1,axiom,( ! [A,B] : ( l1_ami_1(B,A) => l1_struct_0(B) ) ), file(ami_1,l1_ami_1), []). fof(dt_l1_struct_0,axiom,( $true ), file(struct_0,l1_struct_0), []). fof(dt_m1_ami_1,axiom,( ! [A,B] : ( l1_ami_1(B,A) => ! [C] : ( m1_ami_1(C,A,B) => m1_subset_1(C,k7_ami_1(u5_ami_1(A,B))) ) ) ), file(ami_1,m1_ami_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_relset_1,axiom,( $true ), file(relset_1,m1_relset_1), []). fof(dt_m1_scmpds_2,axiom,( ! [A] : ( m1_scmpds_2(A) => m1_subset_1(A,u1_struct_0(k1_scmpds_2)) ) ), file(scmpds_2,m1_scmpds_2), []). fof(dt_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), []). fof(dt_m1_trees_4,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m1_trees_4(C,A,B) => m2_finseq_1(C,A) ) ) ), file(trees_4,m1_trees_4), []). fof(dt_m2_finseq_1,axiom,( ! [A,B] : ( m2_finseq_1(B,A) => ( v1_funct_1(B) & v1_finseq_1(B) & m2_relset_1(B,k5_numbers,A) ) ) ), file(finseq_1,m2_finseq_1), []). fof(dt_m2_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_u1_ami_1,axiom,( ! [A,B] : ( l1_ami_1(B,A) => m1_subset_1(u1_ami_1(A,B),u1_struct_0(B)) ) ), file(ami_1,u1_ami_1), []). fof(dt_u1_struct_0,axiom,( $true ), file(struct_0,u1_struct_0), []). fof(dt_u2_ami_1,axiom,( ! [A,B] : ( l1_ami_1(B,A) => m1_subset_1(u2_ami_1(A,B),k1_zfmisc_1(u1_struct_0(B))) ) ), file(ami_1,u2_ami_1), []). fof(dt_u3_ami_1,axiom,( ! [A,B] : ( l1_ami_1(B,A) => ~ v1_xboole_0(u3_ami_1(A,B)) ) ), file(ami_1,u3_ami_1), []). fof(dt_u4_ami_1,axiom,( ! [A,B] : ( l1_ami_1(B,A) => ( ~ v1_xboole_0(u4_ami_1(A,B)) & m1_subset_1(u4_ami_1(A,B),k1_zfmisc_1(k2_zfmisc_1(u3_ami_1(A,B),k13_finseq_1(k2_xboole_0(k3_tarski(A),u1_struct_0(B)))))) ) ) ), file(ami_1,u4_ami_1), []). fof(dt_u5_ami_1,axiom,( ! [A,B] : ( l1_ami_1(B,A) => ( v1_funct_1(u5_ami_1(A,B)) & v1_funct_2(u5_ami_1(A,B),u1_struct_0(B),k2_xboole_0(A,k2_tarski(u4_ami_1(A,B),u2_ami_1(A,B)))) & m2_relset_1(u5_ami_1(A,B),u1_struct_0(B),k2_xboole_0(A,k2_tarski(u4_ami_1(A,B),u2_ami_1(A,B)))) ) ) ), file(ami_1,u5_ami_1), []). fof(dt_u6_ami_1,axiom,( ! [A,B] : ( l1_ami_1(B,A) => ( v1_funct_1(u6_ami_1(A,B)) & v1_funct_2(u6_ami_1(A,B),u4_ami_1(A,B),k1_funct_2(k4_card_3(u5_ami_1(A,B)),k4_card_3(u5_ami_1(A,B)))) & m2_relset_1(u6_ami_1(A,B),u4_ami_1(A,B),k1_funct_2(k4_card_3(u5_ami_1(A,B)),k4_card_3(u5_ami_1(A,B)))) ) ) ), file(ami_1,u6_ami_1), []). fof(existence_l1_ami_1,axiom,( ! [A] : ? [B] : l1_ami_1(B,A) ), file(ami_1,l1_ami_1), []). fof(existence_l1_struct_0,axiom,( ? [A] : l1_struct_0(A) ), file(struct_0,l1_struct_0), []). fof(existence_m1_ami_1,axiom,( ! [A,B] : ( l1_ami_1(B,A) => ? [C] : m1_ami_1(C,A,B) ) ), file(ami_1,m1_ami_1), []). fof(existence_m1_finseq_1,axiom,( ! [A] : ? [B] : m1_finseq_1(B,A) ), file(finseq_1,m1_finseq_1), []). fof(existence_m1_relset_1,axiom,( ! [A,B] : ? [C] : m1_relset_1(C,A,B) ), file(relset_1,m1_relset_1), []). fof(existence_m1_scmpds_2,axiom,( ? [A] : m1_scmpds_2(A) ), file(scmpds_2,m1_scmpds_2), []). fof(existence_m1_subset_1,axiom,( ! [A] : ? [B] : m1_subset_1(B,A) ), file(subset_1,m1_subset_1), []). fof(existence_m1_trees_4,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ? [C] : m1_trees_4(C,A,B) ) ), file(trees_4,m1_trees_4), []). fof(existence_m2_finseq_1,axiom,( ! [A] : ? [B] : m2_finseq_1(B,A) ), file(finseq_1,m2_finseq_1), []). fof(existence_m2_relset_1,axiom,( ! [A,B] : ? [C] : m2_relset_1(C,A,B) ), file(relset_1,m2_relset_1), []). fof(existence_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ? [C] : m2_subset_1(C,A,B) ) ), file(subset_1,m2_subset_1), []). fof(fc10_ami_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( ~ v1_xboole_0(k7_ami_1(A)) & v1_fraenkel(k7_ami_1(A)) ) ) ), file(ami_1,fc10_ami_1), []). fof(fc1_fraenkel,axiom,( ! [A,B] : v1_fraenkel(k1_funct_2(A,B)) ), file(fraenkel,fc1_fraenkel), []). fof(fc1_int_1,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v1_int_1(k2_xcmplx_0(A,B)) ) ) ), file(int_1,fc1_int_1), []). fof(fc1_numbers,axiom,( ~ v1_xboole_0(k1_numbers) ), file(numbers,fc1_numbers), []). 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_scmpds_2,axiom, ( ~ v3_struct_0(k1_scmpds_2) & v1_ami_1(k1_scmpds_2,k1_tarski(k4_numbers)) & ~ v2_ami_1(k1_scmpds_2,k1_tarski(k4_numbers)) ), file(scmpds_2,fc1_scmpds_2), []). fof(fc1_scmpds_8,axiom,( ! [A,B,C] : ( ( v1_ami_3(A,k1_tarski(k4_numbers),k1_scmpds_2) & v1_scmpds_3(A) & v3_scmpds_4(A) & m1_ami_1(A,k1_tarski(k4_numbers),k1_scmpds_2) & m1_scmpds_2(B) & v1_int_1(C) ) => ( v1_relat_1(k2_scmpds_8(B,C,A)) & v1_funct_1(k2_scmpds_8(B,C,A)) & v1_finset_1(k2_scmpds_8(B,C,A)) & v1_ami_3(k2_scmpds_8(B,C,A),k1_tarski(k4_numbers),k1_scmpds_2) & v1_scmpds_3(k2_scmpds_8(B,C,A)) & v3_scmpds_4(k2_scmpds_8(B,C,A)) ) ) ), file(scmpds_8,fc1_scmpds_8), []). fof(fc2_ami_1,axiom,( ! [A,B] : ( ( ~ v2_ami_1(B,A) & l1_ami_1(B,A) ) => ~ v1_xboole_0(u2_ami_1(A,B)) ) ), file(ami_1,fc2_ami_1), []). fof(fc2_fraenkel,axiom,( ! [A,B] : ( ( v1_finset_1(A) & v1_finset_1(B) ) => ( v1_finset_1(k1_funct_2(A,B)) & v1_fraenkel(k1_funct_2(A,B)) ) ) ), file(fraenkel,fc2_fraenkel), []). fof(fc2_scmpds_2,axiom, ( ~ v3_struct_0(k1_scmpds_2) & v1_ami_1(k1_scmpds_2,k1_tarski(k4_numbers)) & ~ v2_ami_1(k1_scmpds_2,k1_tarski(k4_numbers)) & v5_ami_1(k1_scmpds_2,k1_tarski(k4_numbers)) & v6_ami_1(k1_scmpds_2,k1_tarski(k4_numbers)) & v8_ami_1(k1_scmpds_2,k1_tarski(k4_numbers)) ), file(scmpds_2,fc2_scmpds_2), []). fof(fc2_scmpds_8,axiom,( ! [A,B,C] : ( ( v1_ami_3(A,k1_tarski(k4_numbers),k1_scmpds_2) & v1_scmpds_3(A) & v3_scmpds_5(A) & m1_ami_1(A,k1_tarski(k4_numbers),k1_scmpds_2) & m1_scmpds_2(B) & v1_int_1(C) ) => ( v1_relat_1(k2_scmpds_8(B,C,A)) & v1_funct_1(k2_scmpds_8(B,C,A)) & v1_finset_1(k2_scmpds_8(B,C,A)) & v1_ami_3(k2_scmpds_8(B,C,A),k1_tarski(k4_numbers),k1_scmpds_2) & v1_scmpds_3(k2_scmpds_8(B,C,A)) & v3_scmpds_5(k2_scmpds_8(B,C,A)) ) ) ), file(scmpds_8,fc2_scmpds_8), []). fof(fc2_setfam_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ( ~ v1_xboole_0(k1_tarski(A)) & v1_setfam_1(k1_tarski(A)) ) ) ), file(setfam_1,fc2_setfam_1), []). fof(fc2_wsierp_1,axiom,( ! [A,B] : ( m1_finseq_1(A,k4_numbers) => ( v1_xcmplx_0(k1_funct_1(A,B)) & v1_xreal_0(k1_funct_1(A,B)) & v1_int_1(k1_funct_1(A,B)) & v1_rat_1(k1_funct_1(A,B)) ) ) ), file(wsierp_1,fc2_wsierp_1), []). fof(fc3_ami_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => v1_fraenkel(k4_card_3(A)) ) ), file(ami_1,fc3_ami_1), []). fof(fc3_int_1,axiom,( ! [A] : ( v1_int_1(A) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & v1_int_1(k4_xcmplx_0(A)) ) ) ), file(int_1,fc3_int_1), []). fof(fc3_setfam_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ( ~ v1_xboole_0(k2_tarski(A,B)) & v1_setfam_1(k2_tarski(A,B)) ) ) ), file(setfam_1,fc3_setfam_1), []). fof(fc4_ami_1,axiom,( ! [A,B,C] : ( ( v1_setfam_1(B) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) ) => ( ~ v1_xboole_0(k4_card_3(C)) & v1_fraenkel(k4_card_3(C)) ) ) ), file(ami_1,fc4_ami_1), []). fof(fc4_int_1,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(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)) ) ) ), file(int_1,fc4_int_1), []). fof(fc4_numbers,axiom,( ~ v1_xboole_0(k4_numbers) ), file(numbers,fc4_numbers), []). fof(fc4_scmpds_2,axiom, ( ~ v3_struct_0(k1_scmpds_2) & v1_ami_1(k1_scmpds_2,k1_tarski(k4_numbers)) & ~ v2_ami_1(k1_scmpds_2,k1_tarski(k4_numbers)) & v4_ami_1(k1_scmpds_2,k1_tarski(k4_numbers)) & v5_ami_1(k1_scmpds_2,k1_tarski(k4_numbers)) & v6_ami_1(k1_scmpds_2,k1_tarski(k4_numbers)) & v8_ami_1(k1_scmpds_2,k1_tarski(k4_numbers)) ), file(scmpds_2,fc4_scmpds_2), []). fof(fc4_setfam_1,axiom,( ! [A,B] : ( ( v1_setfam_1(A) & v1_setfam_1(B) ) => v1_setfam_1(k2_xboole_0(A,B)) ) ), file(setfam_1,fc4_setfam_1), []). fof(fc5_int_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & ~ v2_xreal_0(k4_xcmplx_0(A)) & v1_int_1(k4_xcmplx_0(A)) ) ) ), file(int_1,fc5_int_1), []). fof(fc5_scmpds_2,axiom, ( ~ v3_struct_0(k1_scmpds_2) & v1_ami_1(k1_scmpds_2,k1_tarski(k4_numbers)) & ~ v2_ami_1(k1_scmpds_2,k1_tarski(k4_numbers)) & v4_ami_1(k1_scmpds_2,k1_tarski(k4_numbers)) & v5_ami_1(k1_scmpds_2,k1_tarski(k4_numbers)) & v6_ami_1(k1_scmpds_2,k1_tarski(k4_numbers)) & v7_ami_1(k1_scmpds_2,k1_tarski(k4_numbers)) & v8_ami_1(k1_scmpds_2,k1_tarski(k4_numbers)) & v10_ami_1(k1_scmpds_2,k1_tarski(k4_numbers)) ), file(scmpds_2,fc5_scmpds_2), []). fof(fc6_int_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_int_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v1_int_1(k2_xcmplx_0(B,A)) ) ) ), file(int_1,fc6_int_1), []). fof(fc8_int_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_int_1(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)) ) ) ), file(int_1,fc8_int_1), []). fof(fc9_int_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v1_int_1(k6_xcmplx_0(A,B)) ) ) ), file(int_1,fc9_int_1), []). fof(free_g1_ami_1,axiom,( ! [A,B,C,D,E,F,G,H] : ( ( m1_subset_1(C,B) & m1_subset_1(D,k1_zfmisc_1(B)) & ~ v1_xboole_0(E) & ~ v1_xboole_0(F) & m1_subset_1(F,k1_zfmisc_1(k2_zfmisc_1(E,k13_finseq_1(k2_xboole_0(k3_tarski(A),B))))) & v1_funct_1(G) & v1_funct_2(G,B,k2_xboole_0(A,k2_tarski(F,D))) & m1_relset_1(G,B,k2_xboole_0(A,k2_tarski(F,D))) & v1_funct_1(H) & v1_funct_2(H,F,k1_funct_2(k4_card_3(G),k4_card_3(G))) & m1_relset_1(H,F,k1_funct_2(k4_card_3(G),k4_card_3(G))) ) => ! [I,J,K,L,M,N,O,P] : ( g1_ami_1(A,B,C,D,E,F,G,H) = g1_ami_1(I,J,K,L,M,N,O,P) => ( A = I & B = J & C = K & D = L & E = M & F = N & G = O & H = P ) ) ) ), file(ami_1,g1_ami_1), []). fof(idempotence_k2_xboole_0,axiom,( ! [A,B] : k2_xboole_0(A,A) = A ), file(xboole_0,k2_xboole_0), []). fof(involutiveness_k4_xcmplx_0,axiom,( ! [A] : ( v1_xcmplx_0(A) => k4_xcmplx_0(k4_xcmplx_0(A)) = A ) ), file(xcmplx_0,k4_xcmplx_0), []). fof(rc1_ami_1,axiom,( ! [A] : ? [B] : ( l1_ami_1(B,A) & v1_ami_1(B,A) ) ), file(ami_1,rc1_ami_1), []). fof(rc1_fraenkel,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_fraenkel(A) ) ), file(fraenkel,rc1_fraenkel), []). fof(rc1_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc1_funct_1), []). fof(rc1_int_1,axiom,( ? [A] : ( m1_subset_1(A,k1_numbers) & v1_xcmplx_0(A) & v1_xreal_0(A) & v1_int_1(A) ) ), file(int_1,rc1_int_1), []). fof(rc1_scmfsa_4,axiom,( ! [A,B] : ( l1_ami_1(B,A) => ? [C] : ( m1_ami_1(C,A,B) & v1_relat_1(C) & v1_funct_1(C) & v1_finset_1(C) & v1_ami_3(C,A,B) ) ) ), file(scmfsa_4,rc1_scmfsa_4), []). fof(rc1_setfam_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_setfam_1(A) ) ), file(setfam_1,rc1_setfam_1), []). fof(rc2_ami_1,axiom,( ! [A] : ? [B] : ( l1_ami_1(B,A) & ~ v3_struct_0(B) & ~ v2_ami_1(B,A) ) ), file(ami_1,rc2_ami_1), []). fof(rc2_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) ), file(funct_1,rc2_funct_1), []). fof(rc2_int_1,axiom,( ? [A] : v1_int_1(A) ), file(int_1,rc2_int_1), []). fof(rc3_ami_1,axiom,( ! [A] : ( v1_setfam_1(A) => ? [B] : ( l1_ami_1(B,A) & ~ v2_ami_1(B,A) & v4_ami_1(B,A) ) ) ), file(ami_1,rc3_ami_1), []). fof(rc3_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ), file(funct_1,rc3_funct_1), []). fof(rc5_ami_1,axiom,( ! [A] : ? [B] : ( l1_ami_1(B,A) & v1_ami_1(B,A) & v6_ami_1(B,A) ) ), file(ami_1,rc5_ami_1), []). fof(rc6_ami_1,axiom,( ! [A] : ? [B] : ( l1_ami_1(B,A) & ~ v3_struct_0(B) & v1_ami_1(B,A) & ~ v2_ami_1(B,A) & v5_ami_1(B,A) & v6_ami_1(B,A) & v8_ami_1(B,A) ) ), file(ami_1,rc6_ami_1), []). fof(rc7_ami_1,axiom,( ! [A] : ( v1_setfam_1(A) => ? [B] : ( l1_ami_1(B,A) & ~ v3_struct_0(B) & v1_ami_1(B,A) & ~ v2_ami_1(B,A) & v4_ami_1(B,A) & v5_ami_1(B,A) & v6_ami_1(B,A) & v7_ami_1(B,A) & v8_ami_1(B,A) ) ) ), file(ami_1,rc7_ami_1), []). fof(rc8_ami_1,axiom,( ! [A] : ? [B] : ( l1_ami_1(B,A) & v1_ami_1(B,A) & v6_ami_1(B,A) & v10_ami_1(B,A) ) ), file(ami_1,rc8_ami_1), []). fof(rc9_ami_1,axiom,( ! [A] : ? [B] : ( l1_ami_1(B,A) & ~ v3_struct_0(B) & v1_ami_1(B,A) & ~ v2_ami_1(B,A) & v5_ami_1(B,A) & v6_ami_1(B,A) & v8_ami_1(B,A) & v10_ami_1(B,A) ) ), file(ami_1,rc9_ami_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_k1_wsierp_1,axiom,( ! [A,B] : ( ( m1_finseq_1(A,k1_numbers) & m1_subset_1(B,k5_numbers) ) => k1_wsierp_1(A,B) = k1_funct_1(A,B) ) ), file(wsierp_1,k1_wsierp_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_scmpds_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) & m1_scmpds_2(B) ) => k2_scmpds_2(A,B) = k1_funct_1(A,B) ) ), file(scmpds_2,k2_scmpds_2), []). 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_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_k5_numbers,axiom,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), []). fof(redefinition_k6_wsierp_1,axiom,( k6_wsierp_1 = k4_numbers ), file(wsierp_1,k6_wsierp_1), []). fof(redefinition_k7_wsierp_1,axiom,( ! [A] : ( m1_finseq_1(A,k6_wsierp_1) => k7_wsierp_1(A) = k15_rvsum_1(A) ) ), file(wsierp_1,k7_wsierp_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_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_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(s1_scmpds_8__e19_10__scpinvar,axiom,( ! [A,B,C,D,E,F,G,H,I] : ( ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) & v1_ami_3(B,k1_tarski(k4_numbers),k1_scmpds_2) & v1_scmpds_3(B) & v3_scmpds_4(B) & v3_scmpds_5(B) & m1_ami_1(B,k1_tarski(k4_numbers),k1_scmpds_2) & m1_scmpds_2(C) & m1_scmpds_2(D) & m1_scmpds_2(E) & m2_subset_1(F,k1_numbers,k5_numbers) & m2_subset_1(G,k1_numbers,k5_numbers) & m2_subset_1(H,k1_numbers,k5_numbers) & v1_funct_1(I) & v1_funct_2(I,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)),k5_numbers) & m2_relset_1(I,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)),k5_numbers) ) => ( ( ~ r1_xreal_0(k4_card_1(B),0) & ! [J] : ( m1_subset_1(J,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) => ( ( ! [K] : ( m2_subset_1(K,k1_numbers,k5_numbers) => ( ~ r1_xreal_0(K,H) => k2_scmpds_2(k1_scmpds_8(J),k1_scmp_gcd(K)) = k2_scmpds_2(A,k1_scmp_gcd(K)) ) ) & ? [K] : ( m1_trees_4(K,k1_numbers,k6_wsierp_1) & r1_scpisort(K,A,H) & k3_finseq_1(K) = k2_xcmplx_0(k2_scmpds_2(k1_scmpds_8(J),k1_scmp_gcd(G)),F) & k2_scmpds_2(k1_scmpds_8(J),D) = k7_wsierp_1(K) & r1_xreal_0(k2_scmpds_2(k1_scmpds_8(J),k1_scmp_gcd(G)),0) & k2_scmpds_2(k1_scmpds_8(J),E) = k1_nat_1(k1_nat_1(H,1),k3_finseq_1(K)) ) & k8_funct_2(k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)),k5_numbers,I,k1_scmpds_8(J)) = 0 ) => r1_xreal_0(0,k2_scmpds_2(J,k3_scmpds_2(k2_scmpds_2(A,C),G))) ) ) & ! [L] : ( m2_subset_1(L,k1_numbers,k5_numbers) => ( ~ r1_xreal_0(L,H) => k2_scmpds_2(k1_scmpds_8(A),k1_scmp_gcd(L)) = k2_scmpds_2(A,k1_scmp_gcd(L)) ) ) & ? [L] : ( m1_trees_4(L,k1_numbers,k6_wsierp_1) & r1_scpisort(L,A,H) & k3_finseq_1(L) = k2_xcmplx_0(k2_scmpds_2(k1_scmpds_8(A),k1_scmp_gcd(G)),F) & k2_scmpds_2(k1_scmpds_8(A),D) = k7_wsierp_1(L) & r1_xreal_0(k2_scmpds_2(k1_scmpds_8(A),k1_scmp_gcd(G)),0) & k2_scmpds_2(k1_scmpds_8(A),E) = k1_nat_1(k1_nat_1(H,1),k3_finseq_1(L)) ) & ! [J] : ( m1_subset_1(J,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) => ( ( ! [M] : ( m2_subset_1(M,k1_numbers,k5_numbers) => ( ~ r1_xreal_0(M,H) => k2_scmpds_2(k1_scmpds_8(J),k1_scmp_gcd(M)) = k2_scmpds_2(A,k1_scmp_gcd(M)) ) ) & ? [M] : ( m1_trees_4(M,k1_numbers,k6_wsierp_1) & r1_scpisort(M,A,H) & k3_finseq_1(M) = k2_xcmplx_0(k2_scmpds_2(k1_scmpds_8(J),k1_scmp_gcd(G)),F) & k2_scmpds_2(k1_scmpds_8(J),D) = k7_wsierp_1(M) & r1_xreal_0(k2_scmpds_2(k1_scmpds_8(J),k1_scmp_gcd(G)),0) & k2_scmpds_2(k1_scmpds_8(J),E) = k1_nat_1(k1_nat_1(H,1),k3_finseq_1(M)) ) & k2_scmpds_2(J,C) = k2_scmpds_2(A,C) ) => ( r1_xreal_0(0,k2_scmpds_2(J,k3_scmpds_2(k2_scmpds_2(A,C),G))) | ( k2_scmpds_2(k9_scmpds_4(B,J),C) = k2_scmpds_2(J,C) & r1_scmpds_6(B,J) & r2_scmpds_6(B,J) & ~ r1_xreal_0(k8_funct_2(k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)),k5_numbers,I,k1_scmpds_8(J)),k8_funct_2(k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)),k5_numbers,I,k1_scmpds_8(k9_scmpds_4(B,J)))) & ! [N] : ( m2_subset_1(N,k1_numbers,k5_numbers) => ( ~ r1_xreal_0(N,H) => k2_scmpds_2(k1_scmpds_8(k9_scmpds_4(B,J)),k1_scmp_gcd(N)) = k2_scmpds_2(A,k1_scmp_gcd(N)) ) ) & ? [N] : ( m1_trees_4(N,k1_numbers,k6_wsierp_1) & r1_scpisort(N,A,H) & k3_finseq_1(N) = k2_xcmplx_0(k2_scmpds_2(k1_scmpds_8(k9_scmpds_4(B,J)),k1_scmp_gcd(G)),F) & k2_scmpds_2(k1_scmpds_8(k9_scmpds_4(B,J)),D) = k7_wsierp_1(N) & r1_xreal_0(k2_scmpds_2(k1_scmpds_8(k9_scmpds_4(B,J)),k1_scmp_gcd(G)),0) & k2_scmpds_2(k1_scmpds_8(k9_scmpds_4(B,J)),E) = k1_nat_1(k1_nat_1(H,1),k3_finseq_1(N)) ) ) ) ) ) ) => ( ( k8_funct_2(k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)),k5_numbers,I,A) = k8_funct_2(k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)),k5_numbers,I,A) | ( ! [O] : ( m2_subset_1(O,k1_numbers,k5_numbers) => ( ~ r1_xreal_0(O,H) => k2_scmpds_2(A,k1_scmp_gcd(O)) = k2_scmpds_2(A,k1_scmp_gcd(O)) ) ) & ? [O] : ( m1_trees_4(O,k1_numbers,k6_wsierp_1) & r1_scpisort(O,A,H) & k3_finseq_1(O) = k2_xcmplx_0(k2_scmpds_2(A,k1_scmp_gcd(G)),F) & k2_scmpds_2(A,D) = k7_wsierp_1(O) & r1_xreal_0(k2_scmpds_2(A,k1_scmp_gcd(G)),0) & k2_scmpds_2(A,E) = k1_nat_1(k1_nat_1(H,1),k3_finseq_1(O)) ) ) ) & r1_scmpds_6(k2_scmpds_8(C,G,B),A) & r2_scmpds_6(k2_scmpds_8(C,G,B),A) ) ) ) ), file(scpinvar,s1_scmpds_8__e19_10__scpinvar), []). fof(s1_scpinvar__e9_10__scpinvar,axiom,( ! [A,B,C,D,E,F,G,H,I] : ( ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) & v1_ami_3(B,k1_tarski(k4_numbers),k1_scmpds_2) & v1_scmpds_3(B) & v3_scmpds_4(B) & v3_scmpds_5(B) & m1_ami_1(B,k1_tarski(k4_numbers),k1_scmpds_2) & m1_scmpds_2(C) & m1_scmpds_2(D) & m1_scmpds_2(E) & m2_subset_1(F,k1_numbers,k5_numbers) & m2_subset_1(G,k1_numbers,k5_numbers) & m2_subset_1(H,k1_numbers,k5_numbers) & v1_funct_1(I) & v1_funct_2(I,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)),k5_numbers) & m2_relset_1(I,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)),k5_numbers) ) => ( ( ~ r1_xreal_0(k4_card_1(B),0) & ! [J] : ( m1_subset_1(J,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) => ( ( ! [K] : ( m2_subset_1(K,k1_numbers,k5_numbers) => ( ~ r1_xreal_0(K,H) => k2_scmpds_2(k1_scmpds_8(J),k1_scmp_gcd(K)) = k2_scmpds_2(A,k1_scmp_gcd(K)) ) ) & ? [K] : ( m1_trees_4(K,k1_numbers,k6_wsierp_1) & r1_scpisort(K,A,H) & k3_finseq_1(K) = k2_xcmplx_0(k2_scmpds_2(k1_scmpds_8(J),k1_scmp_gcd(G)),F) & k2_scmpds_2(k1_scmpds_8(J),D) = k7_wsierp_1(K) & r1_xreal_0(k2_scmpds_2(k1_scmpds_8(J),k1_scmp_gcd(G)),0) & k2_scmpds_2(k1_scmpds_8(J),E) = k1_nat_1(k1_nat_1(H,1),k3_finseq_1(K)) ) ) => ( k8_funct_2(k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)),k5_numbers,I,k1_scmpds_8(J)) = 0 <=> r1_xreal_0(0,k2_scmpds_2(J,k3_scmpds_2(k2_scmpds_2(A,C),G))) ) ) ) & ! [L] : ( m2_subset_1(L,k1_numbers,k5_numbers) => ( ~ r1_xreal_0(L,H) => k2_scmpds_2(k1_scmpds_8(A),k1_scmp_gcd(L)) = k2_scmpds_2(A,k1_scmp_gcd(L)) ) ) & ? [L] : ( m1_trees_4(L,k1_numbers,k6_wsierp_1) & r1_scpisort(L,A,H) & k3_finseq_1(L) = k2_xcmplx_0(k2_scmpds_2(k1_scmpds_8(A),k1_scmp_gcd(G)),F) & k2_scmpds_2(k1_scmpds_8(A),D) = k7_wsierp_1(L) & r1_xreal_0(k2_scmpds_2(k1_scmpds_8(A),k1_scmp_gcd(G)),0) & k2_scmpds_2(k1_scmpds_8(A),E) = k1_nat_1(k1_nat_1(H,1),k3_finseq_1(L)) ) & ! [J] : ( m1_subset_1(J,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) => ( ( ! [M] : ( m2_subset_1(M,k1_numbers,k5_numbers) => ( ~ r1_xreal_0(M,H) => k2_scmpds_2(k1_scmpds_8(J),k1_scmp_gcd(M)) = k2_scmpds_2(A,k1_scmp_gcd(M)) ) ) & ? [M] : ( m1_trees_4(M,k1_numbers,k6_wsierp_1) & r1_scpisort(M,A,H) & k3_finseq_1(M) = k2_xcmplx_0(k2_scmpds_2(k1_scmpds_8(J),k1_scmp_gcd(G)),F) & k2_scmpds_2(k1_scmpds_8(J),D) = k7_wsierp_1(M) & r1_xreal_0(k2_scmpds_2(k1_scmpds_8(J),k1_scmp_gcd(G)),0) & k2_scmpds_2(k1_scmpds_8(J),E) = k1_nat_1(k1_nat_1(H,1),k3_finseq_1(M)) ) & k2_scmpds_2(J,C) = k2_scmpds_2(A,C) ) => ( r1_xreal_0(0,k2_scmpds_2(J,k3_scmpds_2(k2_scmpds_2(A,C),G))) | ( k2_scmpds_2(k9_scmpds_4(B,J),C) = k2_scmpds_2(J,C) & r1_scmpds_6(B,J) & r2_scmpds_6(B,J) & ~ r1_xreal_0(k8_funct_2(k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)),k5_numbers,I,k1_scmpds_8(J)),k8_funct_2(k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)),k5_numbers,I,k1_scmpds_8(k9_scmpds_4(B,J)))) & ! [N] : ( m2_subset_1(N,k1_numbers,k5_numbers) => ( ~ r1_xreal_0(N,H) => k2_scmpds_2(k1_scmpds_8(k9_scmpds_4(B,J)),k1_scmp_gcd(N)) = k2_scmpds_2(A,k1_scmp_gcd(N)) ) ) & ? [N] : ( m1_trees_4(N,k1_numbers,k6_wsierp_1) & r1_scpisort(N,A,H) & k3_finseq_1(N) = k2_xcmplx_0(k2_scmpds_2(k1_scmpds_8(k9_scmpds_4(B,J)),k1_scmp_gcd(G)),F) & k2_scmpds_2(k1_scmpds_8(k9_scmpds_4(B,J)),D) = k7_wsierp_1(N) & r1_xreal_0(k2_scmpds_2(k1_scmpds_8(k9_scmpds_4(B,J)),k1_scmp_gcd(G)),0) & k2_scmpds_2(k1_scmpds_8(k9_scmpds_4(B,J)),E) = k1_nat_1(k1_nat_1(H,1),k3_finseq_1(N)) ) ) ) ) ) ) => ( k8_funct_2(k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)),k5_numbers,I,k1_scmpds_8(k9_scmpds_4(k2_scmpds_8(C,G,B),A))) = 0 & ! [O] : ( m2_subset_1(O,k1_numbers,k5_numbers) => ( ~ r1_xreal_0(O,H) => k2_scmpds_2(k1_scmpds_8(k9_scmpds_4(k2_scmpds_8(C,G,B),A)),k1_scmp_gcd(O)) = k2_scmpds_2(A,k1_scmp_gcd(O)) ) ) & ? [O] : ( m1_trees_4(O,k1_numbers,k6_wsierp_1) & r1_scpisort(O,A,H) & k3_finseq_1(O) = k2_xcmplx_0(k2_scmpds_2(k1_scmpds_8(k9_scmpds_4(k2_scmpds_8(C,G,B),A)),k1_scmp_gcd(G)),F) & k2_scmpds_2(k1_scmpds_8(k9_scmpds_4(k2_scmpds_8(C,G,B),A)),D) = k7_wsierp_1(O) & r1_xreal_0(k2_scmpds_2(k1_scmpds_8(k9_scmpds_4(k2_scmpds_8(C,G,B),A)),k1_scmp_gcd(G)),0) & k2_scmpds_2(k1_scmpds_8(k9_scmpds_4(k2_scmpds_8(C,G,B),A)),E) = k1_nat_1(k1_nat_1(H,1),k3_finseq_1(O)) ) ) ) ) ), file(scpinvar,s1_scpinvar__e9_10__scpinvar), []). fof(spc1_numerals,axiom, ( v2_xreal_0(1) & m1_subset_1(1,k5_numbers) ), file(numerals,spc1_numerals), []). fof(t102_rvsum_1,axiom,( k15_rvsum_1(k6_finseq_1(k1_numbers)) = 0 ), file(rvsum_1,t102_rvsum_1), []). fof(t10_finseq_2,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v1_finseq_1(C) ) => ( ( k3_finseq_1(B) = A & k3_finseq_1(C) = A & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( r2_hidden(D,k2_finseq_1(A)) => k1_funct_1(B,D) = k1_funct_1(C,D) ) ) ) => B = C ) ) ) ) ), file(finseq_2,t10_finseq_2), []). fof(t148_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ~ r1_xreal_0(A,k6_xcmplx_0(A,1)) ) ), file(xreal_1,t148_xreal_1), []). fof(t19_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ~ ( 0 != A & r1_xreal_0(A,0) ) ) ), file(nat_1,t19_nat_1), []). fof(t1_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k2_xcmplx_0(A,0) = A ) ), file(arithm,t1_arithm), []). fof(t1_numerals,axiom,( m1_subset_1(0,k5_numbers) ), file(numerals,t1_numerals), []). fof(t1_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & v2_xreal_0(A) ) => v2_xreal_0(B) ) ) ) ), file(real,t1_real), []). fof(t1_subset,axiom,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), []). fof(t1_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & r1_xreal_0(B,A) ) => A = B ) ) ) ), file(xreal_1,t1_xreal_1), []). fof(t20_int_1,axiom,( ! [A] : ( v1_int_1(A) => ! [B] : ( v1_int_1(B) => ( ~ r1_xreal_0(B,A) => r1_xreal_0(k2_xcmplx_0(A,1),B) ) ) ) ), file(int_1,t20_int_1), []). 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(t32_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( A = k6_finseq_1(B) <=> k3_finseq_1(A) = 0 ) ) ), file(finseq_1,t32_finseq_1), []). fof(t3_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( r2_hidden(A,k2_finseq_1(B)) <=> ( r1_xreal_0(1,A) & r1_xreal_0(A,B) ) ) ) ) ), file(finseq_1,t3_finseq_1), []). fof(t3_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( r1_xreal_0(A,B) & ~ v3_xreal_0(A) & v3_xreal_0(B) ) ) ) ), file(real,t3_real), []). fof(t3_scpisort,axiom,( ! [A] : ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ? [D] : ( m2_finseq_1(D,k4_numbers) & k3_finseq_1(D) = B & r1_scpisort(D,A,C) ) ) ) ) ), file(scpisort,t3_scpisort), []). fof(t3_subset,axiom,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), []). fof(t4_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k6_xcmplx_0(A,0) = A ) ), file(arithm,t4_arithm), []). 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_scmpds_8,axiom,( ! [A] : ( m1_scmpds_2(A) => ! [B] : ( m1_subset_1(B,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) => k2_scmpds_2(k1_scmpds_8(B),A) = k2_scmpds_2(B,A) ) ) ), file(scmpds_8,t4_scmpds_8), []). 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(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_scmp_gcd,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => k3_scmpds_2(A,B) = k1_scmp_gcd(k1_nat_1(A,B)) ) ) ), file(scmp_gcd,t5_scmp_gcd), []). fof(t5_subset,axiom,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), []). fof(t6_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(A) | v2_xreal_0(B) | v3_xreal_0(A) ) ) ) ) ), file(real,t6_real), []). fof(t7_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_scpinvar,axiom,( ! [A] : ( m1_scmpds_2(A) => ? [B] : ( v1_funct_1(B) & v1_funct_2(B,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)),k5_numbers) & m2_relset_1(B,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)),k5_numbers) & ! [C] : ( m1_subset_1(C,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) => ( ( r1_xreal_0(0,k2_scmpds_2(C,A)) => k8_funct_2(k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)),k5_numbers,B,C) = 0 ) & ( ~ r1_xreal_0(0,k2_scmpds_2(C,A)) => k8_funct_2(k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)),k5_numbers,B,C) = k4_xcmplx_0(k2_scmpds_2(C,A)) ) ) ) ) ) ), file(scpinvar,t7_scpinvar), []). 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), []).