% Mizar problem: t16_scmring2,scmring2,635,50 fof(t16_scmring2,conjecture,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v1_scmring1(A) & l3_vectsp_1(A) ) => ! [B] : ( m1_scmring2(B,A) => ! [C] : ( m1_scmring2(C,A) => ! [D] : ( m1_subset_1(D,k4_card_3(u5_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)))) => ( k1_funct_1(k4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),k7_scmring2(A,B,C),D),k2_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))) = k11_scmring2(A,k6_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),D)) & k3_scmring2(A,k4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),k7_scmring2(A,B,C),D),B) = k1_group_1(A,k3_scmring2(A,D,B),k3_scmring2(A,D,C)) & ! [E] : ( m1_scmring2(E,A) => ( E != B => k3_scmring2(A,k4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),k7_scmring2(A,B,C),D),E) = k3_scmring2(A,D,E) ) ) ) ) ) ) ) ), inference(mizar_bg_added,[status(thm)],[abstractness_v1_ami_1,antisymmetry_r2_hidden,cc1_finseq_5,cc1_fraenkel,cc1_nat_1,cc1_relset_1,cc1_scmring1,cc1_setfam_1,cc2_nat_1,cc3_nat_1,cc5_scmring1,commutativity_k2_tarski,commutativity_k2_xboole_0,commutativity_k4_rlvect_1,connectedness_r1_xreal_0,d14_scmring1,d15_ami_1,d1_scmring2,d5_ami_1,d5_tarski,d6_scmring2,d7_ami_1,dt_g1_ami_1,dt_k10_finseq_1,dt_k10_scmring1,dt_k11_scmring1,dt_k11_scmring2,dt_k12_finseq_1,dt_k12_scmring1,dt_k13_finseq_1,dt_k13_scmring1,dt_k14_scmring1,dt_k15_ami_2,dt_k15_scmring1,dt_k16_scmring1,dt_k1_cqc_lang,dt_k1_fraenkel,dt_k1_funct_1,dt_k1_funct_2,dt_k1_gr_cy_1,dt_k1_group_1,dt_k1_numbers,dt_k1_rlvect_1,dt_k1_scmring1,dt_k1_scmring2,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_ami_1,dt_k2_ami_2,dt_k2_cqc_lang,dt_k2_finseq_4,dt_k2_rlvect_1,dt_k2_scmring1,dt_k2_tarski,dt_k2_xboole_0,dt_k2_zfmisc_1,dt_k3_ami_2,dt_k3_scmring1,dt_k3_scmring2,dt_k3_tarski,dt_k4_ami_1,dt_k4_card_3,dt_k4_rlvect_1,dt_k4_scmring1,dt_k4_tarski,dt_k5_finseq_1,dt_k5_numbers,dt_k5_ordinal2,dt_k5_scmring1,dt_k6_ami_1,dt_k6_rlvect_1,dt_k6_scmring1,dt_k7_scmring1,dt_k7_scmring2,dt_k8_funct_2,dt_k8_scmring1,dt_k9_scmring1,dt_l1_ami_1,dt_l1_group_1,dt_l1_rlvect_1,dt_l1_struct_0,dt_l1_vectsp_1,dt_l2_struct_0,dt_l2_vectsp_1,dt_l3_vectsp_1,dt_m1_finseq_1,dt_m1_fraenkel,dt_m1_relset_1,dt_m1_scmring2,dt_m1_struct_0,dt_m1_subset_1,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_group_1,existence_l1_rlvect_1,existence_l1_struct_0,existence_l1_vectsp_1,existence_l2_struct_0,existence_l2_vectsp_1,existence_l3_vectsp_1,existence_m1_finseq_1,existence_m1_fraenkel,existence_m1_relset_1,existence_m1_scmring2,existence_m1_struct_0,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_relset_1,existence_m2_subset_1,fc1_ami_5,fc1_finseq_5,fc1_fraenkel,fc1_ordinal2,fc1_scmring2,fc1_struct_0,fc1_xboole_0,fc2_ami_1,fc2_finseq_5,fc2_fraenkel,fc2_scmring1,fc2_scmring2,fc2_setfam_1,fc2_xboole_0,fc3_ami_1,fc3_scmring1,fc3_setfam_1,fc3_xboole_0,fc4_ami_1,fc4_scmring1,fc4_setfam_1,free_g1_ami_1,idempotence_k2_xboole_0,rc1_ami_1,rc1_fraenkel,rc1_nat_1,rc1_setfam_1,rc1_xboole_0,rc2_ami_1,rc2_nat_1,rc2_xboole_0,rc3_nat_1,rc3_struct_0,rc4_struct_0,rc5_struct_0,redefinition_k12_finseq_1,redefinition_k12_scmring1,redefinition_k1_fraenkel,redefinition_k2_cqc_lang,redefinition_k2_finseq_4,redefinition_k3_scmring2,redefinition_k4_rlvect_1,redefinition_k5_numbers,redefinition_k6_scmring1,redefinition_k8_funct_2,redefinition_m1_struct_0,redefinition_m2_finseq_1,redefinition_m2_relset_1,redefinition_m2_subset_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,spc0_boole,spc1_boole,spc1_numerals,spc2_boole,spc2_numerals,spc3_boole,spc3_numerals,spc4_boole,spc4_numerals,spc5_boole,spc5_numerals,spc6_boole,spc6_numerals,spc7_boole,spc7_numerals,spc8_boole,spc8_numerals,t10_gr_cy_1,t10_scmring1,t10_scmring2,t11_scmring1,t11_scmring2,t12_scmring2,t14_scmring1,t15_scmring1,t17_scmring1,t1_boole,t1_numerals,t1_real,t1_scmring2,t1_subset,t2_real,t2_subset,t3_real,t3_subset,t4_real,t4_subset,t5_real,t5_subset,t6_boole,t6_real,t7_boole,t7_real,t8_boole,t8_real,t9_scmring2]), [file(scmring2,t16_scmring2)]). 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_finseq_5,axiom,( ! [A] : ( v1_xboole_0(A) => v1_realset1(A) ) ), file(finseq_5,cc1_finseq_5), []). 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_nat_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(nat_1,cc1_nat_1), []). fof(cc1_relset_1,axiom,( ! [A,B,C] : ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) => v1_relat_1(C) ) ), file(relset_1,cc1_relset_1), []). fof(cc1_scmring1,axiom,( ! [A] : ( ~ v1_finset_1(A) => ( ~ v1_xboole_0(A) & ~ v1_realset1(A) ) ) ), file(scmring1,cc1_scmring1), []). 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_nat_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc2_nat_1), []). fof(cc3_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ( v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc3_nat_1), []). fof(cc5_scmring1,axiom,( ! [A] : ( m1_subset_1(A,k2_ami_2) => ( v4_ordinal2(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xcmplx_0(A) ) ) ), file(scmring1,cc5_scmring1), []). 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_k4_rlvect_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & l1_rlvect_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => k4_rlvect_1(A,B,C) = k4_rlvect_1(A,C,B) ) ), file(rlvect_1,k4_rlvect_1), []). fof(connectedness_r1_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( r1_xreal_0(A,B) | r1_xreal_0(B,A) ) ) ), file(xreal_0,r1_xreal_0), []). fof(d14_scmring1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v4_group_1(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v1_scmring1(A) & l3_vectsp_1(A) ) => ! [B] : ( m1_subset_1(B,k1_scmring1(A)) => ! [C] : ( m1_subset_1(C,k4_card_3(k2_scmring1(A))) => ( ( ? [D] : ( m2_subset_1(D,k5_numbers,k2_ami_2) & ? [E] : ( m2_subset_1(E,k5_numbers,k2_ami_2) & B = k4_tarski(1,k2_finseq_4(k2_ami_2,D,E)) ) ) => k15_scmring1(A,B,C) = k4_scmring1(A,k5_scmring1(A,C,k7_scmring1(A,B),k6_scmring1(A,C,k8_scmring1(A,B))),k15_ami_2(k3_scmring1(A,C))) ) & ( ? [D] : ( m2_subset_1(D,k5_numbers,k2_ami_2) & ? [E] : ( m2_subset_1(E,k5_numbers,k2_ami_2) & B = k4_tarski(2,k2_finseq_4(k2_ami_2,D,E)) ) ) => k15_scmring1(A,B,C) = k4_scmring1(A,k5_scmring1(A,C,k7_scmring1(A,B),k4_rlvect_1(A,k6_scmring1(A,C,k7_scmring1(A,B)),k6_scmring1(A,C,k8_scmring1(A,B)))),k15_ami_2(k3_scmring1(A,C))) ) & ( ? [D] : ( m2_subset_1(D,k5_numbers,k2_ami_2) & ? [E] : ( m2_subset_1(E,k5_numbers,k2_ami_2) & B = k4_tarski(3,k2_finseq_4(k2_ami_2,D,E)) ) ) => k15_scmring1(A,B,C) = k4_scmring1(A,k5_scmring1(A,C,k7_scmring1(A,B),k6_rlvect_1(A,k6_scmring1(A,C,k7_scmring1(A,B)),k6_scmring1(A,C,k8_scmring1(A,B)))),k15_ami_2(k3_scmring1(A,C))) ) & ( ? [D] : ( m2_subset_1(D,k5_numbers,k2_ami_2) & ? [E] : ( m2_subset_1(E,k5_numbers,k2_ami_2) & B = k4_tarski(4,k2_finseq_4(k2_ami_2,D,E)) ) ) => k15_scmring1(A,B,C) = k4_scmring1(A,k5_scmring1(A,C,k7_scmring1(A,B),k1_group_1(A,k6_scmring1(A,C,k7_scmring1(A,B)),k6_scmring1(A,C,k8_scmring1(A,B)))),k15_ami_2(k3_scmring1(A,C))) ) & ( ? [D] : ( m2_subset_1(D,k5_numbers,k3_ami_2) & B = k4_tarski(6,k12_finseq_1(k3_ami_2,D)) ) => k15_scmring1(A,B,C) = k4_scmring1(A,C,k9_scmring1(A,B)) ) & ( ? [D] : ( m2_subset_1(D,k5_numbers,k3_ami_2) & ? [E] : ( m2_subset_1(E,k5_numbers,k2_ami_2) & B = k4_tarski(7,k2_finseq_4(k5_numbers,D,E)) ) ) => k15_scmring1(A,B,C) = k4_scmring1(A,C,k2_cqc_lang(k3_ami_2,k6_scmring1(A,C,k11_scmring1(A,B)),k1_rlvect_1(A),k10_scmring1(A,B),k15_ami_2(k3_scmring1(A,C)))) ) & ( ? [D] : ( m2_subset_1(D,k5_numbers,k2_ami_2) & ? [E] : ( m1_subset_1(E,u1_struct_0(A)) & B = k4_tarski(5,k12_scmring1(A,D,E)) ) ) => k15_scmring1(A,B,C) = k4_scmring1(A,k5_scmring1(A,C,k13_scmring1(A,B),k14_scmring1(A,B)),k15_ami_2(k3_scmring1(A,C))) ) & ( ( ! [D] : ( m2_subset_1(D,k5_numbers,k2_ami_2) => ! [E] : ( m2_subset_1(E,k5_numbers,k2_ami_2) => B != k4_tarski(1,k2_finseq_4(k2_ami_2,D,E)) ) ) & ! [D] : ( m2_subset_1(D,k5_numbers,k2_ami_2) => ! [E] : ( m2_subset_1(E,k5_numbers,k2_ami_2) => B != k4_tarski(2,k2_finseq_4(k2_ami_2,D,E)) ) ) & ! [D] : ( m2_subset_1(D,k5_numbers,k2_ami_2) => ! [E] : ( m2_subset_1(E,k5_numbers,k2_ami_2) => B != k4_tarski(3,k2_finseq_4(k2_ami_2,D,E)) ) ) & ! [D] : ( m2_subset_1(D,k5_numbers,k2_ami_2) => ! [E] : ( m2_subset_1(E,k5_numbers,k2_ami_2) => B != k4_tarski(4,k2_finseq_4(k2_ami_2,D,E)) ) ) & ! [D] : ( m2_subset_1(D,k5_numbers,k3_ami_2) => B != k4_tarski(6,k12_finseq_1(k3_ami_2,D)) ) & ! [D] : ( m2_subset_1(D,k5_numbers,k3_ami_2) => ! [E] : ( m2_subset_1(E,k5_numbers,k2_ami_2) => B != k4_tarski(7,k2_finseq_4(k5_numbers,D,E)) ) ) & ! [D] : ( m2_subset_1(D,k5_numbers,k2_ami_2) => ! [E] : ( m1_subset_1(E,u1_struct_0(A)) => B != k4_tarski(5,k12_scmring1(A,D,E)) ) ) ) => k15_scmring1(A,B,C) = C ) ) ) ) ) ), file(scmring1,d14_scmring1), []). 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(d1_scmring2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v1_scmring1(A) & l3_vectsp_1(A) ) => ! [B] : ( ( v1_ami_1(B,k1_tarski(u1_struct_0(A))) & l1_ami_1(B,k1_tarski(u1_struct_0(A))) ) => ( B = k1_scmring2(A) <=> ( u1_struct_0(B) = k5_numbers & u1_ami_1(k1_tarski(u1_struct_0(A)),B) = 0 & u2_ami_1(k1_tarski(u1_struct_0(A)),B) = k3_ami_2 & u3_ami_1(k1_tarski(u1_struct_0(A)),B) = k1_gr_cy_1(8) & u4_ami_1(k1_tarski(u1_struct_0(A)),B) = k1_scmring1(A) & u5_ami_1(k1_tarski(u1_struct_0(A)),B) = k2_scmring1(A) & u6_ami_1(k1_tarski(u1_struct_0(A)),B) = k16_scmring1(A) ) ) ) ) ), file(scmring2,d1_scmring2), []). fof(d5_ami_1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & l1_ami_1(B,A) ) => k2_ami_1(A,B) = u1_ami_1(A,B) ) ), file(ami_1,d5_ami_1), []). fof(d5_tarski,axiom,( ! [A,B] : k4_tarski(A,B) = k2_tarski(k2_tarski(A,B),k1_tarski(A)) ), file(tarski,d5_tarski), []). fof(d6_scmring2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v1_scmring1(A) & l3_vectsp_1(A) ) => ! [B] : ( m1_scmring2(B,A) => ! [C] : ( m1_scmring2(C,A) => k7_scmring2(A,B,C) = k4_tarski(4,k10_finseq_1(B,C)) ) ) ) ), file(scmring2,d6_scmring2), []). fof(d7_ami_1,axiom,( ! [A] : ( v1_setfam_1(A) => ! [B] : ( ( ~ v2_ami_1(B,A) & l1_ami_1(B,A) ) => ! [C] : ( m2_subset_1(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)) => ! [D] : ( m1_subset_1(D,k4_card_3(u5_ami_1(A,B))) => k4_ami_1(A,B,C,D) = k1_funct_1(k8_funct_2(u4_ami_1(A,B),k1_funct_2(k4_card_3(u5_ami_1(A,B)),k4_card_3(u5_ami_1(A,B))),u6_ami_1(A,B),C),D) ) ) ) ) ), file(ami_1,d7_ami_1), []). 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_finseq_1,axiom,( $true ), file(finseq_1,k10_finseq_1), []). fof(dt_k10_scmring1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & m1_subset_1(B,k1_scmring1(A)) ) => m2_subset_1(k10_scmring1(A,B),k5_numbers,k3_ami_2) ) ), file(scmring1,k10_scmring1), []). fof(dt_k11_scmring1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & m1_subset_1(B,k1_scmring1(A)) ) => m2_subset_1(k11_scmring1(A,B),k5_numbers,k2_ami_2) ) ), file(scmring1,k11_scmring1), []). fof(dt_k11_scmring2,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v1_scmring1(A) & l3_vectsp_1(A) & m1_subset_1(B,u2_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))) ) => m1_struct_0(k11_scmring2(A,B),k1_scmring2(A),u2_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))) ) ), file(scmring2,k11_scmring2), []). fof(dt_k12_finseq_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => m2_finseq_1(k12_finseq_1(A,B),A) ) ), file(finseq_1,k12_finseq_1), []). fof(dt_k12_scmring1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & m1_subset_1(B,k2_ami_2) & m1_subset_1(C,u1_struct_0(A)) ) => m2_finseq_1(k12_scmring1(A,B,C),k2_xboole_0(k2_ami_2,u1_struct_0(A))) ) ), file(scmring1,k12_scmring1), []). fof(dt_k13_finseq_1,axiom,( $true ), file(finseq_1,k13_finseq_1), []). fof(dt_k13_scmring1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & m1_subset_1(B,k1_scmring1(A)) ) => m2_subset_1(k13_scmring1(A,B),k5_numbers,k2_ami_2) ) ), file(scmring1,k13_scmring1), []). fof(dt_k14_scmring1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & m1_subset_1(B,k1_scmring1(A)) ) => m1_subset_1(k14_scmring1(A,B),u1_struct_0(A)) ) ), file(scmring1,k14_scmring1), []). fof(dt_k15_ami_2,axiom,( ! [A] : ( m1_subset_1(A,k3_ami_2) => m2_subset_1(k15_ami_2(A),k5_numbers,k3_ami_2) ) ), file(ami_2,k15_ami_2), []). fof(dt_k15_scmring1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v4_group_1(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v1_scmring1(A) & l3_vectsp_1(A) & m1_subset_1(B,k1_scmring1(A)) & m1_subset_1(C,k4_card_3(k2_scmring1(A))) ) => m1_subset_1(k15_scmring1(A,B,C),k4_card_3(k2_scmring1(A))) ) ), file(scmring1,k15_scmring1), []). fof(dt_k16_scmring1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v4_group_1(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v1_scmring1(A) & l3_vectsp_1(A) ) => ( v1_funct_1(k16_scmring1(A)) & v1_funct_2(k16_scmring1(A),k1_scmring1(A),k1_fraenkel(k4_card_3(k2_scmring1(A)),k4_card_3(k2_scmring1(A)))) & m2_relset_1(k16_scmring1(A),k1_scmring1(A),k1_fraenkel(k4_card_3(k2_scmring1(A)),k4_card_3(k2_scmring1(A)))) ) ) ), file(scmring1,k16_scmring1), []). fof(dt_k1_cqc_lang,axiom,( $true ), file(cqc_lang,k1_cqc_lang), []). fof(dt_k1_fraenkel,axiom,( ! [A,B] : ( ~ v1_xboole_0(B) => m1_fraenkel(k1_fraenkel(A,B),A,B) ) ), file(fraenkel,k1_fraenkel), []). 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_gr_cy_1,axiom,( ! [A] : ( v4_ordinal2(A) => ( ~ v1_xboole_0(k1_gr_cy_1(A)) & m1_subset_1(k1_gr_cy_1(A),k1_zfmisc_1(k5_numbers)) ) ) ), file(gr_cy_1,k1_gr_cy_1), []). fof(dt_k1_group_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_group_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k1_group_1(A,B,C),u1_struct_0(A)) ) ), file(group_1,k1_group_1), []). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), []). fof(dt_k1_rlvect_1,axiom,( ! [A] : ( l2_struct_0(A) => m1_subset_1(k1_rlvect_1(A),u1_struct_0(A)) ) ), file(rlvect_1,k1_rlvect_1), []). fof(dt_k1_scmring1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => m1_subset_1(k1_scmring1(A),k1_zfmisc_1(k2_zfmisc_1(k1_gr_cy_1(8),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(u1_struct_0(A))),k5_numbers))))) ) ), file(scmring1,k1_scmring1), []). fof(dt_k1_scmring2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v1_scmring1(A) & l3_vectsp_1(A) ) => ( v1_ami_1(k1_scmring2(A),k1_tarski(u1_struct_0(A))) & l1_ami_1(k1_scmring2(A),k1_tarski(u1_struct_0(A))) ) ) ), file(scmring2,k1_scmring2), []). 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_cqc_lang,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(A) & m1_subset_1(D,A) & m1_subset_1(E,A) ) => m1_subset_1(k2_cqc_lang(A,B,C,D,E),A) ) ), file(cqc_lang,k2_cqc_lang), []). fof(dt_k2_finseq_4,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => m2_finseq_1(k2_finseq_4(A,B,C),A) ) ), file(finseq_4,k2_finseq_4), []). fof(dt_k2_rlvect_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_rlvect_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k2_rlvect_1(A,B,C),u1_struct_0(A)) ) ), file(rlvect_1,k2_rlvect_1), []). fof(dt_k2_scmring1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => ( v1_funct_1(k2_scmring1(A)) & v1_funct_2(k2_scmring1(A),k5_numbers,k2_xboole_0(k1_tarski(u1_struct_0(A)),k2_tarski(k1_scmring1(A),k3_ami_2))) & m2_relset_1(k2_scmring1(A),k5_numbers,k2_xboole_0(k1_tarski(u1_struct_0(A)),k2_tarski(k1_scmring1(A),k3_ami_2))) ) ) ), file(scmring1,k2_scmring1), []). 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_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), []). fof(dt_k3_ami_2,axiom,( m1_subset_1(k3_ami_2,k1_zfmisc_1(k5_numbers)) ), file(ami_2,k3_ami_2), []). fof(dt_k3_scmring1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & m1_subset_1(B,k4_card_3(k2_scmring1(A))) ) => m2_subset_1(k3_scmring1(A,B),k5_numbers,k3_ami_2) ) ), file(scmring1,k3_scmring1), []). fof(dt_k3_scmring2,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v1_scmring1(A) & l3_vectsp_1(A) & m1_subset_1(B,k4_card_3(u5_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)))) & m1_scmring2(C,A) ) => m1_subset_1(k3_scmring2(A,B,C),u1_struct_0(A)) ) ), file(scmring2,k3_scmring2), []). 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_3,axiom,( $true ), file(card_3,k4_card_3), []). fof(dt_k4_rlvect_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & l1_rlvect_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k4_rlvect_1(A,B,C),u1_struct_0(A)) ) ), file(rlvect_1,k4_rlvect_1), []). fof(dt_k4_scmring1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v1_scmring1(A) & l1_struct_0(A) & m1_subset_1(B,k4_card_3(k2_scmring1(A))) & m1_subset_1(C,k3_ami_2) ) => m1_subset_1(k4_scmring1(A,B,C),k4_card_3(k2_scmring1(A))) ) ), file(scmring1,k4_scmring1), []). fof(dt_k4_tarski,axiom,( $true ), file(tarski,k4_tarski), []). fof(dt_k5_finseq_1,axiom,( $true ), file(finseq_1,k5_finseq_1), []). fof(dt_k5_numbers,axiom,( m1_subset_1(k5_numbers,k1_zfmisc_1(k1_numbers)) ), file(numbers,k5_numbers), []). fof(dt_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), []). fof(dt_k5_scmring1,axiom,( ! [A,B,C,D] : ( ( ~ v3_struct_0(A) & v1_scmring1(A) & l1_struct_0(A) & m1_subset_1(B,k4_card_3(k2_scmring1(A))) & m1_subset_1(C,k2_ami_2) & m1_subset_1(D,u1_struct_0(A)) ) => m1_subset_1(k5_scmring1(A,B,C,D),k4_card_3(k2_scmring1(A))) ) ), file(scmring1,k5_scmring1), []). 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_rlvect_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_rlvect_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k6_rlvect_1(A,B,C),u1_struct_0(A)) ) ), file(rlvect_1,k6_rlvect_1), []). fof(dt_k6_scmring1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v1_scmring1(A) & l1_struct_0(A) & m1_subset_1(B,k4_card_3(k2_scmring1(A))) & m1_subset_1(C,k2_ami_2) ) => m1_subset_1(k6_scmring1(A,B,C),u1_struct_0(A)) ) ), file(scmring1,k6_scmring1), []). fof(dt_k7_scmring1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & m1_subset_1(B,k1_scmring1(A)) ) => m2_subset_1(k7_scmring1(A,B),k5_numbers,k2_ami_2) ) ), file(scmring1,k7_scmring1), []). fof(dt_k7_scmring2,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v1_scmring1(A) & l3_vectsp_1(A) & m1_scmring2(B,A) & m1_scmring2(C,A) ) => m2_subset_1(k7_scmring2(A,B,C),k2_zfmisc_1(u3_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(u1_struct_0(A))),u1_struct_0(k1_scmring2(A))))),u4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))) ) ), file(scmring2,k7_scmring2), []). 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_scmring1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & m1_subset_1(B,k1_scmring1(A)) ) => m2_subset_1(k8_scmring1(A,B),k5_numbers,k2_ami_2) ) ), file(scmring1,k8_scmring1), []). fof(dt_k9_scmring1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & m1_subset_1(B,k1_scmring1(A)) ) => m2_subset_1(k9_scmring1(A,B),k5_numbers,k3_ami_2) ) ), file(scmring1,k9_scmring1), []). 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_group_1,axiom,( ! [A] : ( l1_group_1(A) => l1_struct_0(A) ) ), file(group_1,l1_group_1), []). fof(dt_l1_rlvect_1,axiom,( ! [A] : ( l1_rlvect_1(A) => l2_struct_0(A) ) ), file(rlvect_1,l1_rlvect_1), []). fof(dt_l1_struct_0,axiom,( $true ), file(struct_0,l1_struct_0), []). fof(dt_l1_vectsp_1,axiom,( ! [A] : ( l1_vectsp_1(A) => l1_group_1(A) ) ), file(vectsp_1,l1_vectsp_1), []). fof(dt_l2_struct_0,axiom,( ! [A] : ( l2_struct_0(A) => l1_struct_0(A) ) ), file(struct_0,l2_struct_0), []). fof(dt_l2_vectsp_1,axiom,( ! [A] : ( l2_vectsp_1(A) => ( l1_vectsp_1(A) & l2_struct_0(A) ) ) ), file(vectsp_1,l2_vectsp_1), []). fof(dt_l3_vectsp_1,axiom,( ! [A] : ( l3_vectsp_1(A) => ( l1_rlvect_1(A) & l2_vectsp_1(A) ) ) ), file(vectsp_1,l3_vectsp_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_fraenkel,axiom,( ! [A,B,C] : ( m1_fraenkel(C,A,B) => ( ~ v1_xboole_0(C) & v1_fraenkel(C) ) ) ), file(fraenkel,m1_fraenkel), []). fof(dt_m1_relset_1,axiom,( $true ), file(relset_1,m1_relset_1), []). fof(dt_m1_scmring2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v1_scmring1(A) & l3_vectsp_1(A) ) => ! [B] : ( m1_scmring2(B,A) => m1_subset_1(B,u1_struct_0(k1_scmring2(A))) ) ) ), file(scmring2,m1_scmring2), []). 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_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_group_1,axiom,( ? [A] : l1_group_1(A) ), file(group_1,l1_group_1), []). fof(existence_l1_rlvect_1,axiom,( ? [A] : l1_rlvect_1(A) ), file(rlvect_1,l1_rlvect_1), []). fof(existence_l1_struct_0,axiom,( ? [A] : l1_struct_0(A) ), file(struct_0,l1_struct_0), []). fof(existence_l1_vectsp_1,axiom,( ? [A] : l1_vectsp_1(A) ), file(vectsp_1,l1_vectsp_1), []). fof(existence_l2_struct_0,axiom,( ? [A] : l2_struct_0(A) ), file(struct_0,l2_struct_0), []). fof(existence_l2_vectsp_1,axiom,( ? [A] : l2_vectsp_1(A) ), file(vectsp_1,l2_vectsp_1), []). fof(existence_l3_vectsp_1,axiom,( ? [A] : l3_vectsp_1(A) ), file(vectsp_1,l3_vectsp_1), []). fof(existence_m1_finseq_1,axiom,( ! [A] : ? [B] : m1_finseq_1(B,A) ), file(finseq_1,m1_finseq_1), []). fof(existence_m1_fraenkel,axiom,( ! [A,B] : ? [C] : m1_fraenkel(C,A,B) ), file(fraenkel,m1_fraenkel), []). fof(existence_m1_relset_1,axiom,( ! [A,B] : ? [C] : m1_relset_1(C,A,B) ), file(relset_1,m1_relset_1), []). fof(existence_m1_scmring2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v1_scmring1(A) & l3_vectsp_1(A) ) => ? [B] : m1_scmring2(B,A) ) ), file(scmring2,m1_scmring2), []). 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_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(fc1_ami_5,axiom, ( ~ v1_xboole_0(k2_ami_2) & ~ v1_finset_1(k2_ami_2) & v1_membered(k2_ami_2) & v2_membered(k2_ami_2) & v3_membered(k2_ami_2) & v4_membered(k2_ami_2) & v5_membered(k2_ami_2) ), file(ami_5,fc1_ami_5), []). fof(fc1_finseq_5,axiom,( ! [A] : ( ~ v1_xboole_0(k5_finseq_1(A)) & v1_relat_1(k5_finseq_1(A)) & v1_funct_1(k5_finseq_1(A)) & v1_realset1(k5_finseq_1(A)) & v1_finset_1(k5_finseq_1(A)) & v1_finseq_1(k5_finseq_1(A)) ) ), file(finseq_5,fc1_finseq_5), []). fof(fc1_fraenkel,axiom,( ! [A,B] : v1_fraenkel(k1_funct_2(A,B)) ), file(fraenkel,fc1_fraenkel), []). 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_scmring2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v1_scmring1(A) & l3_vectsp_1(A) ) => ( ~ v3_struct_0(k1_scmring2(A)) & v1_ami_1(k1_scmring2(A),k1_tarski(u1_struct_0(A))) & ~ v2_ami_1(k1_scmring2(A),k1_tarski(u1_struct_0(A))) ) ) ), file(scmring2,fc1_scmring2), []). fof(fc1_struct_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => ~ v1_xboole_0(u1_struct_0(A)) ) ), file(struct_0,fc1_struct_0), []). fof(fc1_xboole_0,axiom,( v1_xboole_0(k1_xboole_0) ), file(xboole_0,fc1_xboole_0), []). 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_finseq_5,axiom,( ! [A,B] : ( v1_relat_1(k10_finseq_1(A,B)) & v1_funct_1(k10_finseq_1(A,B)) & ~ v1_realset1(k10_finseq_1(A,B)) & v1_finset_1(k10_finseq_1(A,B)) & v1_finseq_1(k10_finseq_1(A,B)) ) ), file(finseq_5,fc2_finseq_5), []). 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_scmring1,axiom, ( ~ v1_xboole_0(k3_ami_2) & ~ v1_finset_1(k3_ami_2) & ~ v1_realset1(k3_ami_2) & v1_membered(k3_ami_2) & v2_membered(k3_ami_2) & v3_membered(k3_ami_2) & v4_membered(k3_ami_2) & v5_membered(k3_ami_2) ), file(scmring1,fc2_scmring1), []). fof(fc2_scmring2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v1_scmring1(A) & l3_vectsp_1(A) ) => ( ~ v3_struct_0(k1_scmring2(A)) & v1_ami_1(k1_scmring2(A),k1_tarski(u1_struct_0(A))) & ~ v2_ami_1(k1_scmring2(A),k1_tarski(u1_struct_0(A))) & v5_ami_1(k1_scmring2(A),k1_tarski(u1_struct_0(A))) ) ) ), file(scmring2,fc2_scmring2), []). 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_xboole_0,axiom,( ! [A,B] : ( ~ v1_xboole_0(A) => ~ v1_xboole_0(k2_xboole_0(A,B)) ) ), file(xboole_0,fc2_xboole_0), []). 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_scmring1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => ( v1_relat_1(k1_scmring1(A)) & ~ v1_xboole_0(k1_scmring1(A)) & ~ v1_realset1(k1_scmring1(A)) ) ) ), file(scmring1,fc3_scmring1), []). 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_xboole_0,axiom,( ! [A,B] : ( ~ v1_xboole_0(A) => ~ v1_xboole_0(k2_xboole_0(B,A)) ) ), file(xboole_0,fc3_xboole_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_scmring1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & v1_funct_1(B) & v1_funct_2(B,k1_scmring1(A),k1_fraenkel(k4_card_3(k2_scmring1(A)),k4_card_3(k2_scmring1(A)))) & m1_relset_1(B,k1_scmring1(A),k1_fraenkel(k4_card_3(k2_scmring1(A)),k4_card_3(k2_scmring1(A)))) & m1_subset_1(C,k1_scmring1(A)) ) => ( v1_relat_1(k1_funct_1(B,C)) & v1_funct_1(k1_funct_1(B,C)) ) ) ), file(scmring1,fc4_scmring1), []). 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(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(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_nat_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc1_nat_1), []). fof(rc1_setfam_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_setfam_1(A) ) ), file(setfam_1,rc1_setfam_1), []). fof(rc1_xboole_0,axiom,( ? [A] : v1_xboole_0(A) ), file(xboole_0,rc1_xboole_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_nat_1,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(A) & v3_ordinal1(A) ) ), file(nat_1,rc2_nat_1), []). fof(rc2_xboole_0,axiom,( ? [A] : ~ v1_xboole_0(A) ), file(xboole_0,rc2_xboole_0), []). fof(rc3_nat_1,axiom,( ? [A] : ( m1_subset_1(A,k5_numbers) & ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc3_nat_1), []). fof(rc3_struct_0,axiom,( ? [A] : ( l1_struct_0(A) & ~ v3_struct_0(A) ) ), file(struct_0,rc3_struct_0), []). fof(rc4_struct_0,axiom,( ? [A] : ( l2_struct_0(A) & ~ v3_struct_0(A) ) ), file(struct_0,rc4_struct_0), []). fof(rc5_struct_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(B) ) ) ), file(struct_0,rc5_struct_0), []). fof(redefinition_k12_finseq_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => k12_finseq_1(A,B) = k5_finseq_1(B) ) ), file(finseq_1,k12_finseq_1), []). fof(redefinition_k12_scmring1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & m1_subset_1(B,k2_ami_2) & m1_subset_1(C,u1_struct_0(A)) ) => k12_scmring1(A,B,C) = k10_finseq_1(B,C) ) ), file(scmring1,k12_scmring1), []). fof(redefinition_k1_fraenkel,axiom,( ! [A,B] : ( ~ v1_xboole_0(B) => k1_fraenkel(A,B) = k1_funct_2(A,B) ) ), file(fraenkel,k1_fraenkel), []). fof(redefinition_k2_cqc_lang,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(A) & m1_subset_1(D,A) & m1_subset_1(E,A) ) => k2_cqc_lang(A,B,C,D,E) = k1_cqc_lang(B,C,D,E) ) ), file(cqc_lang,k2_cqc_lang), []). fof(redefinition_k2_finseq_4,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) & m1_subset_1(C,A) ) => k2_finseq_4(A,B,C) = k10_finseq_1(B,C) ) ), file(finseq_4,k2_finseq_4), []). fof(redefinition_k3_scmring2,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v1_scmring1(A) & l3_vectsp_1(A) & m1_subset_1(B,k4_card_3(u5_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)))) & m1_scmring2(C,A) ) => k3_scmring2(A,B,C) = k1_funct_1(B,C) ) ), file(scmring2,k3_scmring2), []). fof(redefinition_k4_rlvect_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & l1_rlvect_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => k4_rlvect_1(A,B,C) = k2_rlvect_1(A,B,C) ) ), file(rlvect_1,k4_rlvect_1), []). fof(redefinition_k5_numbers,axiom,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), []). fof(redefinition_k6_scmring1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v1_scmring1(A) & l1_struct_0(A) & m1_subset_1(B,k4_card_3(k2_scmring1(A))) & m1_subset_1(C,k2_ami_2) ) => k6_scmring1(A,B,C) = k1_funct_1(B,C) ) ), file(scmring1,k6_scmring1), []). 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_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(spc0_boole,axiom,( v1_xboole_0(0) ), file(boole,spc0_boole), []). fof(spc1_boole,axiom,( ~ v1_xboole_0(1) ), file(boole,spc1_boole), []). fof(spc1_numerals,axiom, ( v2_xreal_0(1) & m1_subset_1(1,k5_numbers) ), file(numerals,spc1_numerals), []). fof(spc2_boole,axiom,( ~ v1_xboole_0(2) ), file(boole,spc2_boole), []). fof(spc2_numerals,axiom, ( v2_xreal_0(2) & m1_subset_1(2,k5_numbers) ), file(numerals,spc2_numerals), []). fof(spc3_boole,axiom,( ~ v1_xboole_0(3) ), file(boole,spc3_boole), []). fof(spc3_numerals,axiom, ( v2_xreal_0(3) & m1_subset_1(3,k5_numbers) ), file(numerals,spc3_numerals), []). fof(spc4_boole,axiom,( ~ v1_xboole_0(4) ), file(boole,spc4_boole), []). fof(spc4_numerals,axiom, ( v2_xreal_0(4) & m1_subset_1(4,k5_numbers) ), file(numerals,spc4_numerals), []). fof(spc5_boole,axiom,( ~ v1_xboole_0(5) ), file(boole,spc5_boole), []). fof(spc5_numerals,axiom, ( v2_xreal_0(5) & m1_subset_1(5,k5_numbers) ), file(numerals,spc5_numerals), []). fof(spc6_boole,axiom,( ~ v1_xboole_0(6) ), file(boole,spc6_boole), []). fof(spc6_numerals,axiom, ( v2_xreal_0(6) & m1_subset_1(6,k5_numbers) ), file(numerals,spc6_numerals), []). fof(spc7_boole,axiom,( ~ v1_xboole_0(7) ), file(boole,spc7_boole), []). fof(spc7_numerals,axiom, ( v2_xreal_0(7) & m1_subset_1(7,k5_numbers) ), file(numerals,spc7_numerals), []). fof(spc8_boole,axiom,( ~ v1_xboole_0(8) ), file(boole,spc8_boole), []). fof(spc8_numerals,axiom, ( v2_xreal_0(8) & m1_subset_1(8,k5_numbers) ), file(numerals,spc8_numerals), []). fof(t10_gr_cy_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( ~ r1_xreal_0(A,0) => ( r2_hidden(B,k1_gr_cy_1(A)) <=> ~ r1_xreal_0(A,B) ) ) ) ) ), file(gr_cy_1,t10_gr_cy_1), []). fof(t10_scmring1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v1_scmring1(A) & l1_struct_0(A) ) => ! [B] : ( m1_subset_1(B,k4_card_3(k2_scmring1(A))) => ! [C] : ( m2_subset_1(C,k5_numbers,k3_ami_2) => k1_funct_1(k4_scmring1(A,B,C),0) = C ) ) ) ), file(scmring1,t10_scmring1), []). fof(t10_scmring2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v1_scmring1(A) & l3_vectsp_1(A) ) => ! [B] : ( m1_subset_1(B,k4_card_3(u5_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)))) => ! [C] : ( m1_subset_1(C,k4_card_3(k2_scmring1(A))) => ( C = B => k6_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),B) = k3_scmring1(A,C) ) ) ) ) ), file(scmring2,t10_scmring2), []). fof(t11_scmring1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v1_scmring1(A) & l1_struct_0(A) ) => ! [B] : ( m1_subset_1(B,k4_card_3(k2_scmring1(A))) => ! [C] : ( m2_subset_1(C,k5_numbers,k3_ami_2) => ! [D] : ( m2_subset_1(D,k5_numbers,k2_ami_2) => k1_funct_1(k4_scmring1(A,B,C),D) = k1_funct_1(B,D) ) ) ) ) ), file(scmring1,t11_scmring1), []). fof(t11_scmring2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v1_scmring1(A) & l3_vectsp_1(A) ) => ! [B] : ( m1_struct_0(B,k1_scmring2(A),u2_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))) => ! [C] : ( m2_subset_1(C,k5_numbers,k3_ami_2) => ( C = B => k15_ami_2(C) = k11_scmring2(A,B) ) ) ) ) ), file(scmring2,t11_scmring2), []). fof(t12_scmring2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v1_scmring1(A) & l3_vectsp_1(A) ) => ! [B] : ( m1_subset_1(B,k4_card_3(u5_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)))) => ! [C] : ( m2_subset_1(C,k2_zfmisc_1(u3_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(u1_struct_0(A))),u1_struct_0(k1_scmring2(A))))),u4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A))) => ! [D] : ( m1_subset_1(D,k1_scmring1(A)) => ( D = C => ! [E] : ( m1_subset_1(E,k4_card_3(k2_scmring1(A))) => ( E = B => k4_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A),C,B) = k15_scmring1(A,D,E) ) ) ) ) ) ) ) ), file(scmring2,t12_scmring2), []). fof(t14_scmring1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v1_scmring1(A) & l1_struct_0(A) ) => ! [B] : ( m1_subset_1(B,k4_card_3(k2_scmring1(A))) => ! [C] : ( m2_subset_1(C,k5_numbers,k2_ami_2) => ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => k1_funct_1(k5_scmring1(A,B,C,D),C) = D ) ) ) ) ), file(scmring1,t14_scmring1), []). fof(t15_scmring1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v1_scmring1(A) & l1_struct_0(A) ) => ! [B] : ( m1_subset_1(B,k4_card_3(k2_scmring1(A))) => ! [C] : ( m2_subset_1(C,k5_numbers,k2_ami_2) => ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ! [E] : ( m2_subset_1(E,k5_numbers,k2_ami_2) => ( E != C => k1_funct_1(k5_scmring1(A,B,C,D),E) = k1_funct_1(B,E) ) ) ) ) ) ) ), file(scmring1,t15_scmring1), []). fof(t17_scmring1,axiom,( ! [A] : ( m2_subset_1(A,k5_numbers,k1_gr_cy_1(8)) => ! [B] : ( ( ~ v3_struct_0(B) & l1_struct_0(B) ) => ! [C] : ( m1_subset_1(C,k1_scmring1(B)) => ! [D] : ( m2_subset_1(D,k5_numbers,k2_ami_2) => ! [E] : ( m2_subset_1(E,k5_numbers,k2_ami_2) => ( C = k4_tarski(A,k2_finseq_4(k2_ami_2,D,E)) => ( k7_scmring1(B,C) = D & k8_scmring1(B,C) = E ) ) ) ) ) ) ) ), file(scmring1,t17_scmring1), []). 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_scmring2,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & v3_rlvect_1(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v4_group_1(B) & v6_vectsp_1(B) & v7_vectsp_1(B) & v8_vectsp_1(B) & v1_scmring1(B) & l3_vectsp_1(B) ) => ( m1_scmring2(A,B) <=> r2_hidden(A,k2_ami_2) ) ) ), file(scmring2,t1_scmring2), []). fof(t1_subset,axiom,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), []). 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(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(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(t5_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(B) | v3_xreal_0(A) | v2_xreal_0(B) ) ) ) ) ), file(real,t5_real), []). fof(t5_subset,axiom,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), []). fof(t6_boole,axiom,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), []). fof(t6_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(A) | v2_xreal_0(B) | v3_xreal_0(A) ) ) ) ) ), file(real,t6_real), []). fof(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(t9_scmring2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v1_scmring1(A) & l3_vectsp_1(A) ) => k2_ami_1(k1_tarski(u1_struct_0(A)),k1_scmring2(A)) = 0 ) ), file(scmring2,t9_scmring2), []).