% Mizar problem: t94_scmpds_6,scmpds_6,3839,60 fof(t94_scmpds_6,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] : ( v1_int_1(D) => ( ( r1_scmpds_6(B,A) & r2_scmpds_6(B,A) ) => ( r1_xreal_0(k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,C),D)),0) | k9_scmpds_4(k7_scmpds_6(C,D,B),A) = k8_ami_5(k1_tarski(k4_numbers),k1_scmpds_2,k9_scmpds_4(B,A),k12_ami_3(k1_tarski(k4_numbers),k1_scmpds_2,k2_scmpds_3(k1_nat_1(k4_card_1(B),1)))) ) ) ) ) ) ) ), inference(mizar_bg_added,[status(thm)],[abstractness_v1_ami_1,antisymmetry_r2_hidden,cc1_fraenkel,cc1_funct_1,cc1_int_1,cc1_scmfsa_4,cc1_setfam_1,cc1_xreal_0,cc2_funct_1,cc2_int_1,cc2_xreal_0,cc3_int_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,commutativity_k1_nat_1,commutativity_k2_tarski,commutativity_k2_xboole_0,commutativity_k2_xcmplx_0,connectedness_r1_xreal_0,d15_ami_1,d18_ami_1,d19_ami_1,d1_tarski,d20_ami_1,d2_scm_1,d3_scmpds_3,d3_scmpds_6,d8_scmpds_4,d9_scmpds_6,dt_g1_ami_1,dt_k10_ami_1,dt_k11_ami_1,dt_k12_ami_1,dt_k12_ami_3,dt_k13_ami_1,dt_k13_finseq_1,dt_k18_scmpds_2,dt_k19_scmpds_2,dt_k1_card_1,dt_k1_funct_1,dt_k1_funct_2,dt_k1_funct_4,dt_k1_nat_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_scmpds_2,dt_k1_struct_0,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_ami_1,dt_k2_ami_2,dt_k2_scm_1,dt_k2_scmpds_2,dt_k2_scmpds_3,dt_k2_scmpds_4,dt_k2_tarski,dt_k2_xboole_0,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k3_scmpds_2,dt_k3_scmpds_3,dt_k3_tarski,dt_k4_ami_1,dt_k4_card_1,dt_k4_card_3,dt_k4_numbers,dt_k4_scmpds_4,dt_k5_ami_1,dt_k5_numbers,dt_k5_ordinal2,dt_k6_ami_1,dt_k6_scmpds_3,dt_k7_ami_1,dt_k7_relat_1,dt_k7_scmpds_6,dt_k8_ami_1,dt_k8_ami_5,dt_k8_funct_2,dt_k8_scmpds_4,dt_k9_ami_1,dt_k9_scmpds_2,dt_k9_scmpds_4,dt_l1_ami_1,dt_l1_struct_0,dt_m1_ami_1,dt_m1_relset_1,dt_m1_scmpds_2,dt_m1_struct_0,dt_m1_subset_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_relset_1,existence_m1_scmpds_2,existence_m1_struct_0,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,fc10_ami_1,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_scmpds_5,fc18_scmpds_4,fc1_fraenkel,fc1_int_1,fc1_numbers,fc1_scmpds_2,fc26_scmpds_5,fc2_ami_1,fc2_fraenkel,fc2_scmpds_2,fc2_setfam_1,fc3_ami_1,fc3_setfam_1,fc3_xreal_0,fc4_ami_1,fc4_funct_1,fc4_numbers,fc4_scmpds_2,fc4_setfam_1,fc5_scmpds_2,fc5_scmpds_5,fc6_ami_1,fc6_int_1,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,free_g1_ami_1,idempotence_k1_funct_4,idempotence_k2_xboole_0,idempotence_k8_ami_5,l69_scmpds_6,rc1_ami_1,rc1_fraenkel,rc1_funct_1,rc1_int_1,rc1_scmfsa_4,rc1_setfam_1,rc1_xreal_0,rc2_ami_1,rc2_funct_1,rc2_int_1,rc2_xreal_0,rc3_ami_1,rc3_funct_1,rc3_xreal_0,rc4_ami_1,rc4_xreal_0,rc5_ami_1,rc6_ami_1,rc7_ami_1,rc8_ami_1,rc9_ami_1,redefinition_k11_ami_1,redefinition_k13_ami_1,redefinition_k1_nat_1,redefinition_k1_struct_0,redefinition_k2_scmpds_2,redefinition_k4_card_1,redefinition_k5_numbers,redefinition_k8_ami_5,redefinition_k8_funct_2,redefinition_m1_struct_0,redefinition_m2_relset_1,redefinition_m2_subset_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,spc0_boole,spc1_boole,spc1_numerals,t12_funct_4,t14_funct_4,t16_scmfsa6b,t1_arithm,t1_boole,t1_numerals,t1_real,t1_scmpds_6,t1_subset,t20_scmpds_4,t21_scmpds_6,t22_nat_1,t22_scmpds_6,t23_scmpds_4,t24_scmpds_4,t26_funct_4,t26_scmpds_6,t2_real,t2_subset,t30_scmpds_6,t31_scmpds_4,t34_ami_3,t36_ami_3,t36_scmpds_4,t38_scmpds_6,t3_real,t3_subset,t45_scmpds_6,t48_ami_1,t48_scmpds_6,t49_scmpds_2,t4_real,t4_subset,t50_ami_3,t51_ami_1,t52_scmpds_2,t53_scmpds_2,t5_real,t5_subset,t68_scmpds_2,t6_boole,t6_real,t70_scmpds_4,t7_boole,t7_real,t8_boole,t8_real,t8_xreal_1,t9_funct_1]), [file(scmpds_6,t94_scmpds_6)]). 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_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(cc1_xreal_0,axiom,( ! [A] : ( v4_ordinal2(A) => v1_xreal_0(A) ) ), file(xreal_0,cc1_xreal_0), []). fof(cc2_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ) ), file(funct_1,cc2_funct_1), []). fof(cc2_int_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_int_1(A) ) ) ), file(int_1,cc2_int_1), []). fof(cc2_xreal_0,axiom,( ! [A] : ( v1_xreal_0(A) => v1_xcmplx_0(A) ) ), file(xreal_0,cc2_xreal_0), []). fof(cc3_int_1,axiom,( ! [A] : ( v4_ordinal2(A) => v1_int_1(A) ) ), file(int_1,cc3_int_1), []). fof(cc3_xreal_0,axiom,( ! [A] : ( ( v1_xreal_0(A) & v2_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc3_xreal_0), []). fof(cc4_int_1,axiom,( ! [A] : ( v1_int_1(A) => ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(int_1,cc4_int_1), []). fof(cc4_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_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(d15_ami_1,axiom,( ! [A] : ( v1_setfam_1(A) => ! [B] : ( ( ~ v3_struct_0(B) & ~ v2_ami_1(B,A) & v5_ami_1(B,A) & l1_ami_1(B,A) ) => ! [C] : ( m1_subset_1(C,k4_card_3(u5_ami_1(A,B))) => k6_ami_1(A,B,C) = k1_funct_1(C,k2_ami_1(A,B)) ) ) ) ), file(ami_1,d15_ami_1), []). fof(d18_ami_1,axiom,( ! [A] : ( v1_setfam_1(A) => ! [B] : ( ( ~ v3_struct_0(B) & ~ v2_ami_1(B,A) & v5_ami_1(B,A) & v8_ami_1(B,A) & l1_ami_1(B,A) ) => ! [C] : ( m1_subset_1(C,k4_card_3(u5_ami_1(A,B))) => k9_ami_1(A,B,C) = k4_ami_1(A,B,k8_ami_1(A,B,C),C) ) ) ) ), file(ami_1,d18_ami_1), []). fof(d19_ami_1,axiom,( ! [A] : ( v1_setfam_1(A) => ! [B] : ( ( ~ v3_struct_0(B) & ~ v2_ami_1(B,A) & v5_ami_1(B,A) & v8_ami_1(B,A) & l1_ami_1(B,A) ) => ! [C] : ( m1_subset_1(C,k4_card_3(u5_ami_1(A,B))) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k5_numbers,k4_card_3(u5_ami_1(A,B))) & m2_relset_1(D,k5_numbers,k4_card_3(u5_ami_1(A,B))) ) => ( D = k10_ami_1(A,B,C) <=> ( k8_funct_2(k5_numbers,k4_card_3(u5_ami_1(A,B)),D,0) = C & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => k8_funct_2(k5_numbers,k4_card_3(u5_ami_1(A,B)),D,k1_nat_1(E,1)) = k9_ami_1(A,B,k8_funct_2(k5_numbers,k4_card_3(u5_ami_1(A,B)),D,E)) ) ) ) ) ) ) ) ), file(ami_1,d19_ami_1), []). fof(d1_tarski,axiom,( ! [A,B] : ( B = k1_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> C = A ) ) ), file(tarski,d1_tarski), []). fof(d20_ami_1,axiom,( ! [A] : ( v1_setfam_1(A) => ! [B] : ( ( ~ v3_struct_0(B) & ~ v2_ami_1(B,A) & v4_ami_1(B,A) & v5_ami_1(B,A) & v8_ami_1(B,A) & l1_ami_1(B,A) ) => ! [C] : ( m1_subset_1(C,k4_card_3(u5_ami_1(A,B))) => ( v9_ami_1(C,A,B) <=> ? [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) & k8_ami_1(A,B,k11_ami_1(A,B,k10_ami_1(A,B,C),D)) = k5_ami_1(A,B) ) ) ) ) ) ), file(ami_1,d20_ami_1), []). fof(d2_scm_1,axiom,( ! [A] : ( v1_setfam_1(A) => ! [B] : ( ( ~ v3_struct_0(B) & ~ v2_ami_1(B,A) & v4_ami_1(B,A) & v5_ami_1(B,A) & v8_ami_1(B,A) & l1_ami_1(B,A) ) => ! [C] : ( m1_subset_1(C,k4_card_3(u5_ami_1(A,B))) => ( v9_ami_1(C,A,B) => ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( D = k2_scm_1(A,B,C) <=> ( k8_ami_1(A,B,k11_ami_1(A,B,k10_ami_1(A,B,C),D)) = k5_ami_1(A,B) & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( k8_ami_1(A,B,k11_ami_1(A,B,k10_ami_1(A,B,C),E)) = k5_ami_1(A,B) => r1_xreal_0(D,E) ) ) ) ) ) ) ) ) ) ), file(scm_1,d2_scm_1), []). fof(d3_scmpds_3,axiom,( ! [A] : ( m1_struct_0(A,k1_scmpds_2,u2_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m1_struct_0(C,k1_scmpds_2,u2_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)) => ( C = k3_scmpds_3(A,B) <=> ? [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) & A = k2_scmpds_3(D) & C = k2_scmpds_3(k1_nat_1(D,B)) ) ) ) ) ) ), file(scmpds_3,d3_scmpds_3), []). fof(d3_scmpds_6,axiom,( ! [A] : ( ( 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) ) => ! [B] : ( m1_subset_1(B,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) => ( r2_scmpds_6(A,B) <=> v9_ami_1(k8_ami_5(k1_tarski(k4_numbers),k1_scmpds_2,B,k2_scmpds_4(k8_scmpds_4(A))),k1_tarski(k4_numbers),k1_scmpds_2) ) ) ) ), file(scmpds_6,d3_scmpds_6), []). fof(d8_scmpds_4,axiom,( ! [A] : ( ( 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) ) => ! [B] : ( m1_subset_1(B,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) => k9_scmpds_4(A,B) = k1_funct_4(k12_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k8_ami_5(k1_tarski(k4_numbers),k1_scmpds_2,B,k2_scmpds_4(k8_scmpds_4(A)))),k7_relat_1(B,u2_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) ) ) ), file(scmpds_4,d8_scmpds_4), []). fof(d9_scmpds_6,axiom,( ! [A] : ( m1_scmpds_2(A) => ! [B] : ( v1_int_1(B) => ! [C] : ( ( 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) ) => k7_scmpds_6(A,B,C) = k4_scmpds_4(k9_scmpds_2(A,B,k1_nat_1(k4_card_1(C),1)),C) ) ) ) ), file(scmpds_6,d9_scmpds_6), []). 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_k10_ami_1,axiom,( ! [A,B,C] : ( ( v1_setfam_1(A) & ~ v3_struct_0(B) & ~ v2_ami_1(B,A) & v5_ami_1(B,A) & v8_ami_1(B,A) & l1_ami_1(B,A) & m1_subset_1(C,k4_card_3(u5_ami_1(A,B))) ) => ( v1_funct_1(k10_ami_1(A,B,C)) & v1_funct_2(k10_ami_1(A,B,C),k5_numbers,k4_card_3(u5_ami_1(A,B))) & m2_relset_1(k10_ami_1(A,B,C),k5_numbers,k4_card_3(u5_ami_1(A,B))) ) ) ), file(ami_1,k10_ami_1), []). fof(dt_k11_ami_1,axiom,( ! [A,B,C,D] : ( ( v1_setfam_1(A) & ~ v2_ami_1(B,A) & l1_ami_1(B,A) & v1_funct_1(C) & v1_funct_2(C,k5_numbers,k4_card_3(u5_ami_1(A,B))) & m1_relset_1(C,k5_numbers,k4_card_3(u5_ami_1(A,B))) & m1_subset_1(D,k5_numbers) ) => m1_subset_1(k11_ami_1(A,B,C,D),k4_card_3(u5_ami_1(A,B))) ) ), file(ami_1,k11_ami_1), []). fof(dt_k12_ami_1,axiom,( ! [A,B,C] : ( ( v1_setfam_1(A) & ~ v3_struct_0(B) & ~ v2_ami_1(B,A) & v4_ami_1(B,A) & v5_ami_1(B,A) & v8_ami_1(B,A) & l1_ami_1(B,A) & m1_subset_1(C,k4_card_3(u5_ami_1(A,B))) ) => m1_subset_1(k12_ami_1(A,B,C),k4_card_3(u5_ami_1(A,B))) ) ), file(ami_1,k12_ami_1), []). fof(dt_k12_ami_3,axiom,( ! [A,B,C] : ( ( v1_setfam_1(A) & ~ v3_struct_0(B) & ~ v2_ami_1(B,A) & v5_ami_1(B,A) & v8_ami_1(B,A) & l1_ami_1(B,A) & m1_subset_1(C,u2_ami_1(A,B)) ) => m1_ami_1(k12_ami_3(A,B,C),A,B) ) ), file(ami_3,k12_ami_3), []). fof(dt_k13_ami_1,axiom,( ! [A,B,C,D] : ( ( v1_setfam_1(A) & ~ v3_struct_0(B) & ~ v2_ami_1(B,A) & v8_ami_1(B,A) & l1_ami_1(B,A) & m1_subset_1(C,k4_card_3(u5_ami_1(A,B))) & m1_subset_1(D,u2_ami_1(A,B)) ) => m2_subset_1(k13_ami_1(A,B,C,D),k2_zfmisc_1(u3_ami_1(A,B),k13_finseq_1(k2_xboole_0(k3_tarski(A),u1_struct_0(B)))),u4_ami_1(A,B)) ) ), file(ami_1,k13_ami_1), []). fof(dt_k13_finseq_1,axiom,( $true ), file(finseq_1,k13_finseq_1), []). fof(dt_k18_scmpds_2,axiom,( ! [A] : ( m1_subset_1(A,u2_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)) => m1_struct_0(k18_scmpds_2(A),k1_scmpds_2,u2_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)) ) ), file(scmpds_2,k18_scmpds_2), []). fof(dt_k19_scmpds_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) & v1_int_1(B) ) => m1_struct_0(k19_scmpds_2(A,B),k1_scmpds_2,u2_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)) ) ), file(scmpds_2,k19_scmpds_2), []). fof(dt_k1_card_1,axiom,( ! [A] : v1_card_1(k1_card_1(A)) ), file(card_1,k1_card_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_funct_4,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_relat_1(B) & v1_funct_1(B) ) => ( v1_relat_1(k1_funct_4(A,B)) & v1_funct_1(k1_funct_4(A,B)) ) ) ), file(funct_4,k1_funct_4), []). 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_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_struct_0,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & m1_subset_1(B,u1_struct_0(A)) ) => m1_subset_1(k1_struct_0(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ), file(struct_0,k1_struct_0), []). 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_ami_1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & l1_ami_1(B,A) ) => m1_subset_1(k2_ami_1(A,B),u1_struct_0(B)) ) ), file(ami_1,k2_ami_1), []). fof(dt_k2_ami_2,axiom,( m1_subset_1(k2_ami_2,k1_zfmisc_1(k5_numbers)) ), file(ami_2,k2_ami_2), []). fof(dt_k2_scm_1,axiom,( ! [A,B,C] : ( ( v1_setfam_1(A) & ~ v3_struct_0(B) & ~ v2_ami_1(B,A) & v4_ami_1(B,A) & v5_ami_1(B,A) & v8_ami_1(B,A) & l1_ami_1(B,A) & m1_subset_1(C,k4_card_3(u5_ami_1(A,B))) ) => m2_subset_1(k2_scm_1(A,B,C),k1_numbers,k5_numbers) ) ), file(scm_1,k2_scm_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_3,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => m1_struct_0(k2_scmpds_3(A),k1_scmpds_2,u2_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)) ) ), file(scmpds_3,k2_scmpds_3), []). fof(dt_k2_scmpds_4,axiom,( ! [A] : ( ( 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_ami_1(k2_scmpds_4(A),k1_tarski(k4_numbers),k1_scmpds_2) ) ), file(scmpds_4,k2_scmpds_4), []). 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_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_scmpds_3,axiom,( ! [A,B] : ( ( m1_subset_1(A,u2_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)) & m1_subset_1(B,k5_numbers) ) => m1_struct_0(k3_scmpds_3(A,B),k1_scmpds_2,u2_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)) ) ), file(scmpds_3,k3_scmpds_3), []). fof(dt_k3_tarski,axiom,( $true ), file(tarski,k3_tarski), []). fof(dt_k4_ami_1,axiom,( ! [A,B,C,D] : ( ( v1_setfam_1(A) & ~ v2_ami_1(B,A) & l1_ami_1(B,A) & m1_subset_1(C,u4_ami_1(A,B)) & m1_subset_1(D,k4_card_3(u5_ami_1(A,B))) ) => m1_subset_1(k4_ami_1(A,B,C,D),k4_card_3(u5_ami_1(A,B))) ) ), file(ami_1,k4_ami_1), []). 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_scmpds_4,axiom,( ! [A,B] : ( ( m1_subset_1(A,u4_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)) & v1_ami_3(B,k1_tarski(k4_numbers),k1_scmpds_2) & v1_scmpds_3(B) & m1_ami_1(B,k1_tarski(k4_numbers),k1_scmpds_2) ) => ( v1_ami_3(k4_scmpds_4(A,B),k1_tarski(k4_numbers),k1_scmpds_2) & v1_scmpds_3(k4_scmpds_4(A,B)) & m1_ami_1(k4_scmpds_4(A,B),k1_tarski(k4_numbers),k1_scmpds_2) ) ) ), file(scmpds_4,k4_scmpds_4), []). fof(dt_k5_ami_1,axiom,( ! [A,B] : ( ( v1_setfam_1(A) & ~ v2_ami_1(B,A) & v4_ami_1(B,A) & l1_ami_1(B,A) ) => m2_subset_1(k5_ami_1(A,B),k2_zfmisc_1(u3_ami_1(A,B),k13_finseq_1(k2_xboole_0(k3_tarski(A),u1_struct_0(B)))),u4_ami_1(A,B)) ) ), file(ami_1,k5_ami_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_ami_1,axiom,( ! [A,B,C] : ( ( v1_setfam_1(A) & ~ v3_struct_0(B) & ~ v2_ami_1(B,A) & v5_ami_1(B,A) & l1_ami_1(B,A) & m1_subset_1(C,k4_card_3(u5_ami_1(A,B))) ) => m1_struct_0(k6_ami_1(A,B,C),B,u2_ami_1(A,B)) ) ), file(ami_1,k6_ami_1), []). fof(dt_k6_scmpds_3,axiom,( ! [A,B] : ( ( v1_ami_3(A,k1_tarski(k4_numbers),k1_scmpds_2) & m1_ami_1(A,k1_tarski(k4_numbers),k1_scmpds_2) & m1_subset_1(B,k5_numbers) ) => ( v1_ami_3(k6_scmpds_3(A,B),k1_tarski(k4_numbers),k1_scmpds_2) & m1_ami_1(k6_scmpds_3(A,B),k1_tarski(k4_numbers),k1_scmpds_2) ) ) ), file(scmpds_3,k6_scmpds_3), []). fof(dt_k7_ami_1,axiom,( $true ), file(ami_1,k7_ami_1), []). fof(dt_k7_relat_1,axiom,( ! [A,B] : ( v1_relat_1(A) => v1_relat_1(k7_relat_1(A,B)) ) ), file(relat_1,k7_relat_1), []). fof(dt_k7_scmpds_6,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(k7_scmpds_6(A,B,C),k1_tarski(k4_numbers),k1_scmpds_2) & v1_scmpds_3(k7_scmpds_6(A,B,C)) & m1_ami_1(k7_scmpds_6(A,B,C),k1_tarski(k4_numbers),k1_scmpds_2) ) ) ), file(scmpds_6,k7_scmpds_6), []). fof(dt_k8_ami_1,axiom,( ! [A,B,C] : ( ( v1_setfam_1(A) & ~ v3_struct_0(B) & ~ v2_ami_1(B,A) & v5_ami_1(B,A) & v8_ami_1(B,A) & l1_ami_1(B,A) & m1_subset_1(C,k4_card_3(u5_ami_1(A,B))) ) => m2_subset_1(k8_ami_1(A,B,C),k2_zfmisc_1(u3_ami_1(A,B),k13_finseq_1(k2_xboole_0(k3_tarski(A),u1_struct_0(B)))),u4_ami_1(A,B)) ) ), file(ami_1,k8_ami_1), []). fof(dt_k8_ami_5,axiom,( ! [A,B,C,D] : ( ( v1_setfam_1(A) & ~ v2_ami_1(B,A) & l1_ami_1(B,A) & m1_subset_1(C,k4_card_3(u5_ami_1(A,B))) & m1_ami_1(D,A,B) ) => m1_subset_1(k8_ami_5(A,B,C,D),k4_card_3(u5_ami_1(A,B))) ) ), file(ami_5,k8_ami_5), []). 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_k8_scmpds_4,axiom,( ! [A] : ( ( 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) ) => ( v1_ami_3(k8_scmpds_4(A),k1_tarski(k4_numbers),k1_scmpds_2) & v1_scmpds_3(k8_scmpds_4(A)) & m1_ami_1(k8_scmpds_4(A),k1_tarski(k4_numbers),k1_scmpds_2) ) ) ), file(scmpds_4,k8_scmpds_4), []). fof(dt_k9_ami_1,axiom,( ! [A,B,C] : ( ( v1_setfam_1(A) & ~ v3_struct_0(B) & ~ v2_ami_1(B,A) & v5_ami_1(B,A) & v8_ami_1(B,A) & l1_ami_1(B,A) & m1_subset_1(C,k4_card_3(u5_ami_1(A,B))) ) => m1_subset_1(k9_ami_1(A,B,C),k4_card_3(u5_ami_1(A,B))) ) ), file(ami_1,k9_ami_1), []). fof(dt_k9_scmpds_2,axiom,( ! [A,B,C] : ( ( m1_scmpds_2(A) & v1_int_1(B) & v1_int_1(C) ) => m2_subset_1(k9_scmpds_2(A,B,C),k2_zfmisc_1(u3_ami_1(k1_tarski(k4_numbers),k1_scmpds_2),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(k4_numbers)),u1_struct_0(k1_scmpds_2)))),u4_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)) ) ), file(scmpds_2,k9_scmpds_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_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_struct_0,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ! [C] : ( m1_struct_0(C,A,B) => m1_subset_1(C,u1_struct_0(A)) ) ) ), file(struct_0,m1_struct_0), []). fof(dt_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_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_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_struct_0,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ? [C] : m1_struct_0(C,A,B) ) ), file(struct_0,m1_struct_0), []). fof(existence_m1_subset_1,axiom,( ! [A] : ? [B] : m1_subset_1(B,A) ), file(subset_1,m1_subset_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(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_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_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(fc14_scmpds_5,axiom, ( v3_ami_1(k5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2),k1_tarski(k4_numbers),k1_scmpds_2) & v2_scmpds_5(k5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)) ), file(scmpds_5,fc14_scmpds_5), []). fof(fc18_scmpds_4,axiom,( ! [A] : ( ( 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) ) => ( v1_relat_1(k8_scmpds_4(A)) & v1_funct_1(k8_scmpds_4(A)) & v1_finset_1(k8_scmpds_4(A)) & v1_ami_3(k8_scmpds_4(A),k1_tarski(k4_numbers),k1_scmpds_2) & v1_scmpds_3(k8_scmpds_4(A)) & v3_scmpds_4(k8_scmpds_4(A)) ) ) ), file(scmpds_4,fc18_scmpds_4), []). 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_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(fc26_scmpds_5,axiom,( ! [A,B] : ( ( v1_scmpds_5(A) & m1_subset_1(A,u4_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_5(B) & m1_ami_1(B,k1_tarski(k4_numbers),k1_scmpds_2) ) => ( v1_relat_1(k4_scmpds_4(A,B)) & v1_funct_1(k4_scmpds_4(A,B)) & v1_finset_1(k4_scmpds_4(A,B)) & v1_ami_3(k4_scmpds_4(A,B),k1_tarski(k4_numbers),k1_scmpds_2) & v1_scmpds_3(k4_scmpds_4(A,B)) & v3_scmpds_5(k4_scmpds_4(A,B)) ) ) ), file(scmpds_5,fc26_scmpds_5), []). 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_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(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_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(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_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_funct_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( v1_relat_1(k7_relat_1(A,B)) & v1_funct_1(k7_relat_1(A,B)) ) ) ), file(funct_1,fc4_funct_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_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(fc5_scmpds_5,axiom,( ! [A,B,C] : ( ( m1_scmpds_2(A) & v1_int_1(B) & v1_int_1(C) ) => v1_scmpds_5(k9_scmpds_2(A,B,C)) ) ), file(scmpds_5,fc5_scmpds_5), []). fof(fc6_ami_1,axiom,( ! [A,B] : ( ( v1_setfam_1(A) & ~ v2_ami_1(B,A) & v4_ami_1(B,A) & l1_ami_1(B,A) ) => v3_ami_1(k5_ami_1(A,B),A,B) ) ), file(ami_1,fc6_ami_1), []). fof(fc6_int_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_int_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v1_int_1(k2_xcmplx_0(B,A)) ) ) ), file(int_1,fc6_int_1), []). fof(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_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(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_k1_funct_4,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_relat_1(B) & v1_funct_1(B) ) => k1_funct_4(A,A) = A ) ), file(funct_4,k1_funct_4), []). fof(idempotence_k2_xboole_0,axiom,( ! [A,B] : k2_xboole_0(A,A) = A ), file(xboole_0,k2_xboole_0), []). fof(idempotence_k8_ami_5,axiom,( ! [A,B,C,D] : ( ( v1_setfam_1(A) & ~ v2_ami_1(B,A) & l1_ami_1(B,A) & m1_subset_1(C,k4_card_3(u5_ami_1(A,B))) & m1_ami_1(D,A,B) ) => k8_ami_5(A,B,C,C) = C ) ), file(ami_5,k8_ami_5), []). fof(l69_scmpds_6,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k2_zfmisc_1(u3_ami_1(k1_tarski(k4_numbers),k1_scmpds_2),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(k4_numbers)),u1_struct_0(k1_scmpds_2)))),u4_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)) => ! [C] : ( m1_subset_1(C,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) => ! [D] : ( ( v1_ami_3(D,k1_tarski(k4_numbers),k1_scmpds_2) & v1_scmpds_3(D) & m1_ami_1(D,k1_tarski(k4_numbers),k1_scmpds_2) ) => r1_tarski(k6_scmpds_3(k8_scmpds_4(D),1),k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k8_ami_5(k1_tarski(k4_numbers),k1_scmpds_2,C,k2_scmpds_4(k8_scmpds_4(k4_scmpds_4(B,D))))),A)) ) ) ) ) ), file(scmpds_6,l69_scmpds_6), []). 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(rc1_xreal_0,axiom,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(xreal_0,rc1_xreal_0), []). 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(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_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(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_ami_1,axiom,( ! [A,B] : ( ( v1_setfam_1(A) & ~ v2_ami_1(B,A) & v4_ami_1(B,A) & l1_ami_1(B,A) ) => ? [C] : ( m1_subset_1(C,u4_ami_1(A,B)) & v3_ami_1(C,A,B) ) ) ), file(ami_1,rc4_ami_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_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_k11_ami_1,axiom,( ! [A,B,C,D] : ( ( v1_setfam_1(A) & ~ v2_ami_1(B,A) & l1_ami_1(B,A) & v1_funct_1(C) & v1_funct_2(C,k5_numbers,k4_card_3(u5_ami_1(A,B))) & m1_relset_1(C,k5_numbers,k4_card_3(u5_ami_1(A,B))) & m1_subset_1(D,k5_numbers) ) => k11_ami_1(A,B,C,D) = k1_funct_1(C,D) ) ), file(ami_1,k11_ami_1), []). fof(redefinition_k13_ami_1,axiom,( ! [A,B,C,D] : ( ( v1_setfam_1(A) & ~ v3_struct_0(B) & ~ v2_ami_1(B,A) & v8_ami_1(B,A) & l1_ami_1(B,A) & m1_subset_1(C,k4_card_3(u5_ami_1(A,B))) & m1_subset_1(D,u2_ami_1(A,B)) ) => k13_ami_1(A,B,C,D) = k1_funct_1(C,D) ) ), file(ami_1,k13_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_struct_0,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & m1_subset_1(B,u1_struct_0(A)) ) => k1_struct_0(A,B) = k1_tarski(B) ) ), file(struct_0,k1_struct_0), []). 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_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_k8_ami_5,axiom,( ! [A,B,C,D] : ( ( v1_setfam_1(A) & ~ v2_ami_1(B,A) & l1_ami_1(B,A) & m1_subset_1(C,k4_card_3(u5_ami_1(A,B))) & m1_ami_1(D,A,B) ) => k8_ami_5(A,B,C,D) = k1_funct_4(C,D) ) ), file(ami_5,k8_ami_5), []). 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_struct_0,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ! [C] : ( m1_struct_0(C,A,B) <=> m1_subset_1(C,B) ) ) ), file(struct_0,m1_struct_0), []). 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(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(t12_funct_4,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( ~ r2_hidden(A,k1_relat_1(B)) => k1_funct_1(k1_funct_4(C,B),A) = k1_funct_1(C,A) ) ) ) ), file(funct_4,t12_funct_4), []). fof(t14_funct_4,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( r2_hidden(A,k1_relat_1(B)) => k1_funct_1(k1_funct_4(C,B),A) = k1_funct_1(B,A) ) ) ) ), file(funct_4,t14_funct_4), []). fof(t16_scmfsa6b,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_setfam_1(A) ) => ! [B] : ( ( ~ v3_struct_0(B) & ~ v2_ami_1(B,A) & v4_ami_1(B,A) & v5_ami_1(B,A) & v8_ami_1(B,A) & l1_ami_1(B,A) ) => ! [C] : ( m1_subset_1(C,k4_card_3(u5_ami_1(A,B))) => ( v9_ami_1(C,A,B) => k12_ami_1(A,B,C) = k11_ami_1(A,B,k10_ami_1(A,B,C),k2_scm_1(A,B,C)) ) ) ) ) ), file(scmfsa6b,t16_scmfsa6b), []). 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_scmpds_6,axiom,( ! [A] : ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) => k1_relat_1(k7_relat_1(A,u2_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) = u2_ami_1(k1_tarski(k4_numbers),k1_scmpds_2) ) ), file(scmpds_6,t1_scmpds_6), []). fof(t1_subset,axiom,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), []). fof(t20_scmpds_4,axiom,( ! [A] : ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) => ! [B] : ~ ( r2_hidden(B,k1_relat_1(A)) & ~ m1_scmpds_2(B) & B != k2_ami_1(k1_tarski(k4_numbers),k1_scmpds_2) & ~ m1_struct_0(B,k1_scmpds_2,u2_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)) ) ) ), file(scmpds_4,t20_scmpds_4), []). fof(t21_scmpds_6,axiom,( ! [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) & m1_ami_1(B,k1_tarski(k4_numbers),k1_scmpds_2) ) => k6_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k8_ami_5(k1_tarski(k4_numbers),k1_scmpds_2,A,k2_scmpds_4(B))) = k2_scmpds_3(0) ) ) ), file(scmpds_6,t21_scmpds_6), []). fof(t22_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ~ ( A != 0 & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => A != k1_nat_1(B,1) ) ) ) ), file(nat_1,t22_nat_1), []). fof(t22_scmpds_6,axiom,( ! [A] : ( m2_subset_1(A,k2_zfmisc_1(u3_ami_1(k1_tarski(k4_numbers),k1_scmpds_2),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(k4_numbers)),u1_struct_0(k1_scmpds_2)))),u4_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)) => ! [B] : ( m1_subset_1(B,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) => ! [C] : ( ( 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) ) => k8_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k8_ami_5(k1_tarski(k4_numbers),k1_scmpds_2,B,k2_scmpds_4(k8_scmpds_4(k4_scmpds_4(A,C))))) = A ) ) ) ), file(scmpds_6,t22_scmpds_6), []). fof(t23_scmpds_4,axiom,( ! [A] : ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) => ! [B] : ( m1_subset_1(B,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) => ( ! [C] : ( m1_scmpds_2(C) => k2_scmpds_2(A,C) = k2_scmpds_2(B,C) ) <=> k7_relat_1(A,k2_ami_2) = k7_relat_1(B,k2_ami_2) ) ) ) ), file(scmpds_4,t23_scmpds_4), []). fof(t24_scmpds_4,axiom,( ! [A] : ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) => ! [B] : ( m1_subset_1(B,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) => ( r1_funct_7(A,B,u2_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)) => k7_relat_1(A,k2_ami_2) = k7_relat_1(B,k2_ami_2) ) ) ) ), file(scmpds_4,t24_scmpds_4), []). fof(t26_funct_4,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => r1_tarski(A,k1_funct_4(B,A)) ) ) ), file(funct_4,t26_funct_4), []). fof(t26_scmpds_6,axiom,( ! [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) & m1_ami_1(B,k1_tarski(k4_numbers),k1_scmpds_2) ) => ! [C] : ( ( 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) ) => ! [D] : ( m1_struct_0(D,k1_scmpds_2,u2_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)) => ! [E] : ( m1_struct_0(E,k1_scmpds_2,u2_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)) => k13_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k9_scmpds_4(B,A),D) = k13_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k8_ami_5(k1_tarski(k4_numbers),k1_scmpds_2,k9_scmpds_4(C,A),k12_ami_3(k1_tarski(k4_numbers),k1_scmpds_2,E)),D) ) ) ) ) ) ), file(scmpds_6,t26_scmpds_6), []). 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(t30_scmpds_6,axiom,( ! [A] : ( m1_scmpds_2(A) => ! [B] : ( v1_int_1(B) => ! [C] : ( v1_int_1(C) => k9_scmpds_2(A,B,C) != k5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2) ) ) ) ), file(scmpds_6,t30_scmpds_6), []). fof(t31_scmpds_4,axiom,( ! [A] : ( ( 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) ) => ! [B] : ( m1_scmpds_2(B) => ~ r2_hidden(B,k1_relat_1(k2_scmpds_4(A))) ) ) ), file(scmpds_4,t31_scmpds_4), []). fof(t34_ami_3,axiom,( ! [A] : ( v1_setfam_1(A) => ! [B] : ( ( ~ v3_struct_0(B) & ~ v2_ami_1(B,A) & v5_ami_1(B,A) & v8_ami_1(B,A) & l1_ami_1(B,A) ) => ! [C] : ( m1_struct_0(C,B,u2_ami_1(A,B)) => k1_relat_1(k12_ami_3(A,B,C)) = k1_struct_0(B,k2_ami_1(A,B)) ) ) ) ), file(ami_3,t34_ami_3), []). fof(t36_ami_3,axiom,( ! [A] : ( v1_setfam_1(A) => ! [B] : ( ( ~ v2_ami_1(B,A) & l1_ami_1(B,A) ) => ! [C] : ( m1_subset_1(C,k4_card_3(u5_ami_1(A,B))) => k1_relat_1(C) = u1_struct_0(B) ) ) ) ), file(ami_3,t36_ami_3), []). fof(t36_scmpds_4,axiom,( ! [A] : ( ( 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) ) => ! [B] : ( ( v1_ami_3(B,k1_tarski(k4_numbers),k1_scmpds_2) & v1_scmpds_3(B) & m1_ami_1(B,k1_tarski(k4_numbers),k1_scmpds_2) ) => ! [C] : ( m1_subset_1(C,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) => r1_funct_7(k8_ami_5(k1_tarski(k4_numbers),k1_scmpds_2,C,k2_scmpds_4(A)),k8_ami_5(k1_tarski(k4_numbers),k1_scmpds_2,C,k2_scmpds_4(B)),u2_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)) ) ) ) ), file(scmpds_4,t36_scmpds_4), []). fof(t38_scmpds_6,axiom,( ! [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) & m1_ami_1(B,k1_tarski(k4_numbers),k1_scmpds_2) ) => ! [C] : ( ( 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) ) => ( r1_scmpds_6(B,A) <=> r1_scmpds_6(B,k8_ami_5(k1_tarski(k4_numbers),k1_scmpds_2,A,k2_scmpds_4(C))) ) ) ) ) ), file(scmpds_6,t38_scmpds_6), []). 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_scmpds_6,axiom,( ! [A] : ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) => ! [B] : ( m1_subset_1(B,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) => ! [C] : ( ( v1_ami_3(C,k1_tarski(k4_numbers),k1_scmpds_2) & v1_scmpds_3(C) & v3_scmpds_4(C) & m1_ami_1(C,k1_tarski(k4_numbers),k1_scmpds_2) ) => ( ( r1_tarski(k2_scmpds_4(k8_scmpds_4(C)),A) & r1_scmpds_6(C,A) ) => ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( ( r1_tarski(k6_scmpds_3(k8_scmpds_4(C),D),B) & k6_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,B) = k2_scmpds_3(D) & k7_relat_1(A,k2_ami_2) = k7_relat_1(B,k2_ami_2) ) => ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( k3_scmpds_3(k6_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),E)),D) = k6_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,B),E)) & k8_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),E)) = k8_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,B),E)) & k7_relat_1(k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),E),k2_ami_2) = k7_relat_1(k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,B),E),k2_ami_2) ) ) ) ) ) ) ) ) ), file(scmpds_6,t45_scmpds_6), []). fof(t48_ami_1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & ~ v2_ami_1(B,A) & v5_ami_1(B,A) & v8_ami_1(B,A) & l1_ami_1(B,A) ) => ( v10_ami_1(B,A) => ! [C] : ( m1_struct_0(C,B,u2_ami_1(A,B)) => k2_ami_1(A,B) != C ) ) ) ), file(ami_1,t48_ami_1), []). fof(t48_scmpds_6,axiom,( ! [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_5(B) & m1_ami_1(B,k1_tarski(k4_numbers),k1_scmpds_2) ) => ( ( r1_scmpds_6(B,A) & r2_scmpds_6(B,A) ) => k6_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k9_scmpds_4(B,A)) = k2_scmpds_3(k4_card_1(B)) ) ) ) ), file(scmpds_6,t48_scmpds_6), []). fof(t49_scmpds_2,axiom,( ! [A] : ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) => ! [B] : ( m1_scmpds_2(B) => r2_hidden(B,k1_relat_1(A)) ) ) ), file(scmpds_2,t49_scmpds_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(t50_ami_3,axiom,( ! [A] : ( v1_setfam_1(A) => ! [B] : ( ( ~ v3_struct_0(B) & ~ v2_ami_1(B,A) & v5_ami_1(B,A) & v8_ami_1(B,A) & l1_ami_1(B,A) ) => ! [C] : ( m1_struct_0(C,B,u2_ami_1(A,B)) => k1_funct_1(k12_ami_3(A,B,C),k2_ami_1(A,B)) = C ) ) ) ), file(ami_3,t50_ami_3), []). fof(t51_ami_1,axiom,( ! [A] : ( v1_setfam_1(A) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( ( ~ v3_struct_0(C) & ~ v2_ami_1(C,A) & v5_ami_1(C,A) & v8_ami_1(C,A) & l1_ami_1(C,A) ) => ! [D] : ( m1_subset_1(D,k4_card_3(u5_ami_1(A,C))) => ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => k11_ami_1(A,C,k10_ami_1(A,C,D),k1_nat_1(B,E)) = k11_ami_1(A,C,k10_ami_1(A,C,k11_ami_1(A,C,k10_ami_1(A,C,D),B)),E) ) ) ) ) ) ), file(ami_1,t51_ami_1), []). fof(t52_scmpds_2,axiom,( ! [A] : ( m1_scmpds_2(A) => A != k2_ami_1(k1_tarski(k4_numbers),k1_scmpds_2) ) ), file(scmpds_2,t52_scmpds_2), []). fof(t53_scmpds_2,axiom,( ! [A] : ( m1_struct_0(A,k1_scmpds_2,u2_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)) => ! [B] : ( m1_scmpds_2(B) => A != B ) ) ), file(scmpds_2,t53_scmpds_2), []). 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(t68_scmpds_2,axiom,( ! [A] : ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))) => ! [B] : ( v1_int_1(B) => ! [C] : ( v1_int_1(C) => ! [D] : ( m1_scmpds_2(D) => ! [E] : ( m1_scmpds_2(E) => ( ( r1_xreal_0(k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,D),B)),0) => k1_funct_1(k4_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k9_scmpds_2(D,B,C),A),k2_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)) = k19_scmpds_2(A,C) ) & ( ~ r1_xreal_0(k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,D),B)),0) => k1_funct_1(k4_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k9_scmpds_2(D,B,C),A),k2_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)) = k18_scmpds_2(k6_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A)) ) & k2_scmpds_2(k4_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k9_scmpds_2(D,B,C),A),E) = k2_scmpds_2(A,E) ) ) ) ) ) ) ), file(scmpds_2,t68_scmpds_2), []). fof(t6_boole,axiom,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), []). fof(t6_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(A) | v2_xreal_0(B) | v3_xreal_0(A) ) ) ) ) ), file(real,t6_real), []). fof(t70_scmpds_4,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k18_scmpds_2(k2_scmpds_3(A)) = k2_scmpds_3(k1_nat_1(A,1)) ) ), file(scmpds_4,t70_scmpds_4), []). fof(t7_boole,axiom,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), []). fof(t7_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,B) & ~ v2_xreal_0(A) & ~ v3_xreal_0(B) ) ) ) ), file(real,t7_real), []). fof(t8_boole,axiom,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), []). fof(t8_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,B) & ~ v3_xreal_0(B) & ~ v2_xreal_0(A) ) ) ) ), file(real,t8_real), []). fof(t8_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( r1_xreal_0(A,B) <=> r1_xreal_0(k2_xcmplx_0(A,C),k2_xcmplx_0(B,C)) ) ) ) ) ), file(xreal_1,t8_xreal_1), []). fof(t9_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( ( k1_relat_1(A) = k1_relat_1(B) & ! [C] : ( r2_hidden(C,k1_relat_1(A)) => k1_funct_1(A,C) = k1_funct_1(B,C) ) ) => A = B ) ) ) ), file(funct_1,t9_funct_1), []).