% Mizar problem: t53_sprect_3,sprect_3,1298,32 fof(t53_sprect_3,conjecture,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_sprect_1(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => k3_goboard2(A) = k6_matrix_2(u1_struct_0(k15_euclid(2)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,4),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,3),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,2)) ) ), inference(mizar_bg_added,[status(thm)],[abstractness_v1_metric_1,abstractness_v1_pre_topc,antisymmetry_r2_hidden,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_finseq_5,cc1_membered,cc1_relset_1,cc1_sprect_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc2_sprect_1,cc2_xreal_0,cc3_arytm_3,cc3_membered,cc3_xreal_0,cc4_membered,cc4_xreal_0,cc5_sprect_1,cc5_xreal_0,cc6_membered,cc6_sprect_1,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,cc9_membered,commutativity_k1_nat_1,commutativity_k2_tarski,commutativity_k2_xboole_0,commutativity_k2_xcmplx_0,connectedness_r1_xreal_0,d16_euclid,d1_enumset1,d1_euclid,d1_finseq_6,d1_goboard1,d1_goboard2,d2_enumset1,d2_goboard2,d2_tarski,d3_goboard1,d3_goboard2,d3_tarski,d4_goboard1,d5_tarski,d7_euclid,d7_finseq_1,d8_euclid,dt_g1_metric_1,dt_g1_pre_topc,dt_k10_finseq_1,dt_k11_finseq_1,dt_k13_euclid,dt_k13_finseq_1,dt_k14_euclid,dt_k15_euclid,dt_k1_card_1,dt_k1_enumset1,dt_k1_euclid,dt_k1_finseq_1,dt_k1_funct_1,dt_k1_goboard1,dt_k1_goboard2,dt_k1_matrix_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k21_euclid,dt_k22_euclid,dt_k22_pscomp_1,dt_k23_euclid,dt_k23_pscomp_1,dt_k24_pscomp_1,dt_k25_pscomp_1,dt_k2_enumset1,dt_k2_finseq_1,dt_k2_finseq_4,dt_k2_goboard1,dt_k2_goboard2,dt_k2_matrix_1,dt_k2_relat_1,dt_k2_tarski,dt_k2_xboole_0,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k30_pscomp_1,dt_k31_pscomp_1,dt_k32_pscomp_1,dt_k33_pscomp_1,dt_k34_pscomp_1,dt_k35_pscomp_1,dt_k36_pscomp_1,dt_k37_pscomp_1,dt_k3_finseq_1,dt_k3_finseq_2,dt_k3_finseq_4,dt_k3_goboard1,dt_k3_goboard2,dt_k3_matrix_1,dt_k3_matrix_2,dt_k4_card_1,dt_k4_finseq_1,dt_k4_finseq_2,dt_k4_finseq_4,dt_k4_tarski,dt_k5_numbers,dt_k5_ordinal2,dt_k5_pcomps_1,dt_k5_topreal1,dt_k6_matrix_2,dt_k7_finseq_1,dt_k8_finseq_1,dt_l1_metric_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_finseq_1,dt_m1_finseq_2,dt_m1_matrix_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_metric_1,dt_u1_pre_topc,dt_u1_struct_0,existence_l1_metric_1,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_finseq_1,existence_m1_finseq_2,existence_m1_matrix_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_relset_1,existence_m2_subset_1,fc10_membered,fc10_xreal_0,fc11_membered,fc11_sprect_1,fc11_xreal_0,fc12_membered,fc12_xreal_0,fc13_membered,fc14_membered,fc15_membered,fc16_membered,fc17_membered,fc18_membered,fc19_membered,fc1_euclid,fc1_goboard2,fc1_sprect_1,fc1_sprect_3,fc1_struct_0,fc20_membered,fc21_membered,fc22_membered,fc23_membered,fc24_membered,fc25_membered,fc26_membered,fc2_euclid,fc2_finseq_5,fc2_goboard2,fc2_membered,fc2_sprect_1,fc3_goboard2,fc3_sprect_1,fc3_sprect_2,fc3_xreal_0,fc4_goboard2,fc5_goboard2,fc5_membered,fc6_membered,fc7_membered,fc7_xreal_0,fc8_membered,fc8_xreal_0,fc9_membered,fc9_xreal_0,free_g1_metric_1,free_g1_pre_topc,idempotence_k2_xboole_0,rc1_arytm_3,rc1_membered,rc1_sprect_1,rc1_sprect_3,rc1_xreal_0,rc2_sprect_1,rc2_xreal_0,rc3_struct_0,rc3_xreal_0,rc4_sprect_1,rc4_xreal_0,rc5_struct_0,redefinition_k1_goboard1,redefinition_k1_nat_1,redefinition_k2_finseq_1,redefinition_k2_finseq_4,redefinition_k3_finseq_1,redefinition_k3_finseq_2,redefinition_k3_finseq_4,redefinition_k4_card_1,redefinition_k4_finseq_1,redefinition_k5_numbers,redefinition_k6_matrix_2,redefinition_k8_finseq_1,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,t105_pscomp_1,t115_pscomp_1,t12_sprect_3,t12_xboole_1,t147_finseq_2,t148_finseq_2,t1_arithm,t1_boole,t1_numerals,t1_real,t1_subset,t27_finseq_3,t29_finseq_3,t2_real,t2_subset,t33_zfmisc_1,t35_finseq_1,t3_matrix_2,t3_real,t3_subset,t44_finseq_1,t4_finseq_1,t4_real,t4_subset,t55_sprect_2,t57_euclid,t5_real,t5_subset,t61_finseq_1,t61_sprect_2,t62_finseq_1,t6_boole,t6_cqc_the1,t6_matrix_2,t6_real,t76_card_2,t7_boole,t7_real,t85_pscomp_1,t86_finseq_1,t8_boole,t8_real,t90_sprect_1,t91_sprect_1,t92_sprect_1,t93_sprect_1,t94_sprect_1,t95_pscomp_1]), [file(sprect_3,t53_sprect_3)]). fof(abstractness_v1_metric_1,axiom,( ! [A] : ( l1_metric_1(A) => ( v1_metric_1(A) => A = g1_metric_1(u1_struct_0(A),u1_metric_1(A)) ) ) ), file(metric_1,v1_metric_1), []). fof(abstractness_v1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ( v1_pre_topc(A) => A = g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) ) ) ), file(pre_topc,v1_pre_topc), []). fof(antisymmetry_r2_hidden,axiom,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), []). fof(cc10_membered,axiom,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,A) => v1_xcmplx_0(B) ) ) ), file(membered,cc10_membered), []). fof(cc11_membered,axiom,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) ) ) ) ), file(membered,cc11_membered), []). fof(cc12_membered,axiom,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_rat_1(B) ) ) ) ), file(membered,cc12_membered), []). fof(cc13_membered,axiom,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc13_membered), []). fof(cc14_membered,axiom,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v4_ordinal2(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc14_membered), []). fof(cc15_membered,axiom,( ! [A] : ( v1_xboole_0(A) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc15_membered), []). fof(cc16_membered,axiom,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_membered(B) ) ) ), file(membered,cc16_membered), []). fof(cc17_membered,axiom,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) ) ) ) ), file(membered,cc17_membered), []). fof(cc18_membered,axiom,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) ) ) ) ), file(membered,cc18_membered), []). fof(cc19_membered,axiom,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) ) ) ) ), file(membered,cc19_membered), []). fof(cc1_arytm_3,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_ordinal1(B) & v2_ordinal1(B) & v3_ordinal1(B) ) ) ) ), file(arytm_3,cc1_arytm_3), []). fof(cc1_finseq_5,axiom,( ! [A] : ( v1_xboole_0(A) => v1_realset1(A) ) ), file(finseq_5,cc1_finseq_5), []). fof(cc1_membered,axiom,( ! [A] : ( v5_membered(A) => v4_membered(A) ) ), file(membered,cc1_membered), []). 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_sprect_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & ~ v5_seqm_3(A) ) => ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & ~ v1_realset1(A) ) ) ), file(sprect_1,cc1_sprect_1), []). fof(cc1_xreal_0,axiom,( ! [A] : ( v4_ordinal2(A) => v1_xreal_0(A) ) ), file(xreal_0,cc1_xreal_0), []). fof(cc20_membered,axiom,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) & v5_membered(B) ) ) ) ), file(membered,cc20_membered), []). fof(cc2_arytm_3,axiom,( ! [A] : ( ( v1_xboole_0(A) & v3_ordinal1(A) ) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ) ), file(arytm_3,cc2_arytm_3), []). fof(cc2_membered,axiom,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), []). fof(cc2_sprect_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_realset1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v5_seqm_3(A) ) ) ), file(sprect_1,cc2_sprect_1), []). fof(cc2_xreal_0,axiom,( ! [A] : ( v1_xreal_0(A) => v1_xcmplx_0(A) ) ), file(xreal_0,cc2_xreal_0), []). fof(cc3_arytm_3,axiom,( ! [A] : ( m1_subset_1(A,k5_ordinal2) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ) ), file(arytm_3,cc3_arytm_3), []). fof(cc3_membered,axiom,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), []). 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_membered,axiom,( ! [A] : ( v2_membered(A) => v1_membered(A) ) ), file(membered,cc4_membered), []). 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_sprect_1,axiom,( ! [A] : ( m1_finseq_1(A,u1_struct_0(k15_euclid(2))) => ( v1_sprect_1(A) => ~ v5_seqm_3(A) ) ) ), file(sprect_1,cc5_sprect_1), []). 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_membered,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( v1_membered(A) & v2_membered(A) ) ) ), file(membered,cc6_membered), []). fof(cc6_sprect_1,axiom,( ! [A] : ( m1_finseq_1(A,u1_struct_0(k15_euclid(2))) => ( ( ~ v1_xboole_0(A) & v1_sprect_1(A) ) => ( ~ v1_xboole_0(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & v2_goboard5(A) ) ) ) ), file(sprect_1,cc6_sprect_1), []). 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(cc9_membered,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k5_numbers)) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc9_membered), []). fof(commutativity_k1_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k1_nat_1(A,B) = k1_nat_1(B,A) ) ), file(nat_1,k1_nat_1), []). fof(commutativity_k2_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(d16_euclid,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => k23_euclid(A,B) = k10_finseq_1(A,B) ) ) ), file(euclid,d16_euclid), []). fof(d1_enumset1,axiom,( ! [A,B,C,D] : ( D = k1_enumset1(A,B,C) <=> ! [E] : ( r2_hidden(E,D) <=> ~ ( E != A & E != B & E != C ) ) ) ), file(enumset1,d1_enumset1), []). fof(d1_euclid,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k1_euclid(A) = k4_finseq_2(A,k1_numbers) ) ), file(euclid,d1_euclid), []). fof(d1_finseq_6,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m2_finseq_1(B,A) => ( v1_finseq_6(B,A) <=> k4_finseq_4(k5_numbers,A,B,1) = k4_finseq_4(k5_numbers,A,B,k3_finseq_1(B)) ) ) ) ), file(finseq_6,d1_finseq_6), []). fof(d1_goboard1,axiom,( ! [A] : ( m2_finseq_1(A,k1_numbers) => ( v1_goboard1(A) <=> ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ~ ( r2_hidden(B,k4_finseq_1(A)) & r2_hidden(C,k4_finseq_1(A)) & ~ r1_xreal_0(C,B) & r1_xreal_0(k1_goboard1(A,C),k1_goboard1(A,B)) ) ) ) ) ) ), file(goboard1,d1_goboard1), []). fof(d1_goboard2,axiom,( ! [A] : ( ( v1_goboard1(A) & m2_finseq_1(A,k1_numbers) ) => ! [B] : ( ( v1_goboard1(B) & m2_finseq_1(B,k1_numbers) ) => ~ ( A != k1_xboole_0 & B != k1_xboole_0 & ~ ! [C] : ( ( v1_matrix_1(C) & m2_finseq_1(C,k3_finseq_2(u1_struct_0(k15_euclid(2)))) ) => ( C = k1_goboard2(A,B) <=> ( k3_finseq_1(C) = k3_finseq_1(A) & k1_matrix_1(C) = k3_finseq_1(B) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( r2_hidden(k4_tarski(D,E),k2_matrix_1(C)) => k3_matrix_1(u1_struct_0(k15_euclid(2)),C,D,E) = k23_euclid(k1_goboard1(A,D),k1_goboard1(B,E)) ) ) ) ) ) ) ) ) ) ), file(goboard2,d1_goboard2), []). fof(d2_enumset1,axiom,( ! [A,B,C,D,E] : ( E = k2_enumset1(A,B,C,D) <=> ! [F] : ( r2_hidden(F,E) <=> ~ ( F != A & F != B & F != C & F != D ) ) ) ), file(enumset1,d2_enumset1), []). fof(d2_goboard2,axiom,( ! [A] : ( m2_finseq_1(A,k1_numbers) => ! [B] : ( ( v1_goboard1(B) & m2_finseq_1(B,k1_numbers) ) => ( B = k2_goboard2(A) <=> ( k2_relat_1(B) = k2_relat_1(A) & k3_finseq_1(B) = k4_card_1(k2_relat_1(A)) ) ) ) ) ), file(goboard2,d2_goboard2), []). fof(d2_tarski,axiom,( ! [A,B,C] : ( C = k2_tarski(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( D = A | D = B ) ) ) ), file(tarski,d2_tarski), []). fof(d3_goboard1,axiom,( ! [A] : ( m2_finseq_1(A,u1_struct_0(k15_euclid(2))) => ! [B] : ( m2_finseq_1(B,k1_numbers) => ( B = k2_goboard1(A) <=> ( k3_finseq_1(B) = k3_finseq_1(A) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( r2_hidden(C,k4_finseq_1(B)) => k1_goboard1(B,C) = k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,C)) ) ) ) ) ) ) ), file(goboard1,d3_goboard1), []). fof(d3_goboard2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => k3_goboard2(A) = k1_goboard2(k2_goboard2(k2_goboard1(A)),k2_goboard2(k3_goboard1(A))) ) ), file(goboard2,d3_goboard2), []). fof(d3_tarski,axiom,( ! [A,B] : ( r1_tarski(A,B) <=> ! [C] : ( r2_hidden(C,A) => r2_hidden(C,B) ) ) ), file(tarski,d3_tarski), []). fof(d4_goboard1,axiom,( ! [A] : ( m2_finseq_1(A,u1_struct_0(k15_euclid(2))) => ! [B] : ( m2_finseq_1(B,k1_numbers) => ( B = k3_goboard1(A) <=> ( k3_finseq_1(B) = k3_finseq_1(A) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( r2_hidden(C,k4_finseq_1(B)) => k1_goboard1(B,C) = k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,C)) ) ) ) ) ) ) ), file(goboard1,d4_goboard1), []). fof(d5_tarski,axiom,( ! [A,B] : k4_tarski(A,B) = k2_tarski(k2_tarski(A,B),k1_tarski(A)) ), file(tarski,d5_tarski), []). fof(d7_euclid,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k14_euclid(A) = g1_metric_1(k1_euclid(A),k13_euclid(A)) ) ), file(euclid,d7_euclid), []). fof(d7_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v1_finseq_1(C) ) => ( C = k7_finseq_1(A,B) <=> ( k4_finseq_1(C) = k2_finseq_1(k1_nat_1(k3_finseq_1(A),k3_finseq_1(B))) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( r2_hidden(D,k4_finseq_1(A)) => k1_funct_1(C,D) = k1_funct_1(A,D) ) ) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( r2_hidden(D,k4_finseq_1(B)) => k1_funct_1(C,k1_nat_1(k3_finseq_1(A),D)) = k1_funct_1(B,D) ) ) ) ) ) ) ) ), file(finseq_1,d7_finseq_1), []). fof(d8_euclid,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k15_euclid(A) = k5_pcomps_1(k14_euclid(A)) ) ), file(euclid,d8_euclid), []). fof(dt_g1_metric_1,axiom,( ! [A,B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers) & m1_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) ) => ( v1_metric_1(g1_metric_1(A,B)) & l1_metric_1(g1_metric_1(A,B)) ) ) ), file(metric_1,g1_metric_1), []). fof(dt_g1_pre_topc,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) => ( v1_pre_topc(g1_pre_topc(A,B)) & l1_pre_topc(g1_pre_topc(A,B)) ) ) ), file(pre_topc,g1_pre_topc), []). fof(dt_k10_finseq_1,axiom,( $true ), file(finseq_1,k10_finseq_1), []). fof(dt_k11_finseq_1,axiom,( $true ), file(finseq_1,k11_finseq_1), []). fof(dt_k13_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_funct_1(k13_euclid(A)) & v1_funct_2(k13_euclid(A),k2_zfmisc_1(k1_euclid(A),k1_euclid(A)),k1_numbers) & m2_relset_1(k13_euclid(A),k2_zfmisc_1(k1_euclid(A),k1_euclid(A)),k1_numbers) ) ) ), file(euclid,k13_euclid), []). fof(dt_k13_finseq_1,axiom,( $true ), file(finseq_1,k13_finseq_1), []). fof(dt_k14_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_metric_1(k14_euclid(A)) & v6_metric_1(k14_euclid(A)) & v7_metric_1(k14_euclid(A)) & v8_metric_1(k14_euclid(A)) & v9_metric_1(k14_euclid(A)) & l1_metric_1(k14_euclid(A)) ) ) ), file(euclid,k14_euclid), []). fof(dt_k15_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_pre_topc(k15_euclid(A)) & v2_pre_topc(k15_euclid(A)) & l1_pre_topc(k15_euclid(A)) ) ) ), file(euclid,k15_euclid), []). fof(dt_k1_card_1,axiom,( ! [A] : v1_card_1(k1_card_1(A)) ), file(card_1,k1_card_1), []). fof(dt_k1_enumset1,axiom,( $true ), file(enumset1,k1_enumset1), []). fof(dt_k1_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( ~ v1_xboole_0(k1_euclid(A)) & m1_finseq_2(k1_euclid(A),k1_numbers) ) ) ), file(euclid,k1_euclid), []). fof(dt_k1_finseq_1,axiom,( $true ), file(finseq_1,k1_finseq_1), []). fof(dt_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), []). fof(dt_k1_goboard1,axiom,( ! [A,B] : ( ( m1_finseq_1(A,k1_numbers) & m1_subset_1(B,k5_numbers) ) => m1_subset_1(k1_goboard1(A,B),k1_numbers) ) ), file(goboard1,k1_goboard1), []). fof(dt_k1_goboard2,axiom,( ! [A,B] : ( ( v1_goboard1(A) & m1_finseq_1(A,k1_numbers) & v1_goboard1(B) & m1_finseq_1(B,k1_numbers) ) => ( v1_matrix_1(k1_goboard2(A,B)) & m2_finseq_1(k1_goboard2(A,B),k3_finseq_2(u1_struct_0(k15_euclid(2)))) ) ) ), file(goboard2,k1_goboard2), []). fof(dt_k1_matrix_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & v1_matrix_1(A) ) => m2_subset_1(k1_matrix_1(A),k1_numbers,k5_numbers) ) ), file(matrix_1,k1_matrix_1), []). fof(dt_k1_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k1_nat_1(A,B),k1_numbers,k5_numbers) ) ), file(nat_1,k1_nat_1), []). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), []). fof(dt_k1_relat_1,axiom,( $true ), file(relat_1,k1_relat_1), []). fof(dt_k1_tarski,axiom,( $true ), file(tarski,k1_tarski), []). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), []). fof(dt_k1_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), []). fof(dt_k21_euclid,axiom,( ! [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) => m1_subset_1(k21_euclid(A),k1_numbers) ) ), file(euclid,k21_euclid), []). fof(dt_k22_euclid,axiom,( ! [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) => m1_subset_1(k22_euclid(A),k1_numbers) ) ), file(euclid,k22_euclid), []). fof(dt_k22_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => m1_subset_1(k22_pscomp_1(A),u1_struct_0(k15_euclid(2))) ) ), file(pscomp_1,k22_pscomp_1), []). fof(dt_k23_euclid,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => m1_subset_1(k23_euclid(A,B),u1_struct_0(k15_euclid(2))) ) ), file(euclid,k23_euclid), []). fof(dt_k23_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => m1_subset_1(k23_pscomp_1(A),u1_struct_0(k15_euclid(2))) ) ), file(pscomp_1,k23_pscomp_1), []). fof(dt_k24_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => m1_subset_1(k24_pscomp_1(A),u1_struct_0(k15_euclid(2))) ) ), file(pscomp_1,k24_pscomp_1), []). fof(dt_k25_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => m1_subset_1(k25_pscomp_1(A),u1_struct_0(k15_euclid(2))) ) ), file(pscomp_1,k25_pscomp_1), []). fof(dt_k2_enumset1,axiom,( $true ), file(enumset1,k2_enumset1), []). fof(dt_k2_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => m1_subset_1(k2_finseq_1(A),k1_zfmisc_1(k5_numbers)) ) ), file(finseq_1,k2_finseq_1), []). fof(dt_k2_finseq_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_goboard1,axiom,( ! [A] : ( m1_finseq_1(A,u1_struct_0(k15_euclid(2))) => m2_finseq_1(k2_goboard1(A),k1_numbers) ) ), file(goboard1,k2_goboard1), []). fof(dt_k2_goboard2,axiom,( ! [A] : ( m1_finseq_1(A,k1_numbers) => ( v1_goboard1(k2_goboard2(A)) & m2_finseq_1(k2_goboard2(A),k1_numbers) ) ) ), file(goboard2,k2_goboard2), []). fof(dt_k2_matrix_1,axiom,( $true ), file(matrix_1,k2_matrix_1), []). fof(dt_k2_relat_1,axiom,( $true ), file(relat_1,k2_relat_1), []). fof(dt_k2_tarski,axiom,( $true ), file(tarski,k2_tarski), []). fof(dt_k2_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_k30_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => m1_subset_1(k30_pscomp_1(A),u1_struct_0(k15_euclid(2))) ) ), file(pscomp_1,k30_pscomp_1), []). fof(dt_k31_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => m1_subset_1(k31_pscomp_1(A),u1_struct_0(k15_euclid(2))) ) ), file(pscomp_1,k31_pscomp_1), []). fof(dt_k32_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => m1_subset_1(k32_pscomp_1(A),u1_struct_0(k15_euclid(2))) ) ), file(pscomp_1,k32_pscomp_1), []). fof(dt_k33_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => m1_subset_1(k33_pscomp_1(A),u1_struct_0(k15_euclid(2))) ) ), file(pscomp_1,k33_pscomp_1), []). fof(dt_k34_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => m1_subset_1(k34_pscomp_1(A),u1_struct_0(k15_euclid(2))) ) ), file(pscomp_1,k34_pscomp_1), []). fof(dt_k35_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => m1_subset_1(k35_pscomp_1(A),u1_struct_0(k15_euclid(2))) ) ), file(pscomp_1,k35_pscomp_1), []). fof(dt_k36_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => m1_subset_1(k36_pscomp_1(A),u1_struct_0(k15_euclid(2))) ) ), file(pscomp_1,k36_pscomp_1), []). fof(dt_k37_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => m1_subset_1(k37_pscomp_1(A),u1_struct_0(k15_euclid(2))) ) ), file(pscomp_1,k37_pscomp_1), []). fof(dt_k3_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => m2_subset_1(k3_finseq_1(A),k1_numbers,k5_numbers) ) ), file(finseq_1,k3_finseq_1), []). fof(dt_k3_finseq_2,axiom,( ! [A] : ( ~ v1_xboole_0(k3_finseq_2(A)) & m1_finseq_2(k3_finseq_2(A),A) ) ), file(finseq_2,k3_finseq_2), []). fof(dt_k3_finseq_4,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) & m1_subset_1(C,A) & m1_subset_1(D,A) ) => m2_finseq_1(k3_finseq_4(A,B,C,D),A) ) ), file(finseq_4,k3_finseq_4), []). fof(dt_k3_goboard1,axiom,( ! [A] : ( m1_finseq_1(A,u1_struct_0(k15_euclid(2))) => m2_finseq_1(k3_goboard1(A),k1_numbers) ) ), file(goboard1,k3_goboard1), []). fof(dt_k3_goboard2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & m1_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ( v1_matrix_1(k3_goboard2(A)) & m2_finseq_1(k3_goboard2(A),k3_finseq_2(u1_struct_0(k15_euclid(2)))) ) ) ), file(goboard2,k3_goboard2), []). fof(dt_k3_matrix_1,axiom,( ! [A,B,C,D] : ( ( v1_matrix_1(B) & m1_finseq_1(B,k3_finseq_2(A)) & m1_subset_1(C,k5_numbers) & m1_subset_1(D,k5_numbers) ) => m1_subset_1(k3_matrix_1(A,B,C,D),A) ) ), file(matrix_1,k3_matrix_1), []). fof(dt_k3_matrix_2,axiom,( ! [A,B,C,D] : ( v1_relat_1(k3_matrix_2(A,B,C,D)) & v1_funct_1(k3_matrix_2(A,B,C,D)) & v1_finseq_1(k3_matrix_2(A,B,C,D)) & v1_matrix_1(k3_matrix_2(A,B,C,D)) ) ), file(matrix_2,k3_matrix_2), []). 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_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => m1_subset_1(k4_finseq_1(A),k1_zfmisc_1(k5_numbers)) ) ), file(finseq_1,k4_finseq_1), []). fof(dt_k4_finseq_2,axiom,( ! [A,B] : ( v4_ordinal2(A) => m1_finseq_2(k4_finseq_2(A,B),B) ) ), file(finseq_2,k4_finseq_2), []). fof(dt_k4_finseq_4,axiom,( ! [A,B,C,D] : ( ( v1_funct_1(C) & m1_relset_1(C,A,B) ) => m1_subset_1(k4_finseq_4(A,B,C,D),B) ) ), file(finseq_4,k4_finseq_4), []). fof(dt_k4_tarski,axiom,( $true ), file(tarski,k4_tarski), []). 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_pcomps_1,axiom,( ! [A] : ( l1_metric_1(A) => l1_pre_topc(k5_pcomps_1(A)) ) ), file(pcomps_1,k5_pcomps_1), []). fof(dt_k5_topreal1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_finseq_1(B,u1_struct_0(k15_euclid(A))) ) => m1_subset_1(k5_topreal1(A,B),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ), file(topreal1,k5_topreal1), []). fof(dt_k6_matrix_2,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) & m1_subset_1(C,A) & m1_subset_1(D,A) & m1_subset_1(E,A) ) => m1_matrix_1(k6_matrix_2(A,B,C,D,E),A,2,2) ) ), file(matrix_2,k6_matrix_2), []). fof(dt_k7_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( v1_relat_1(k7_finseq_1(A,B)) & v1_funct_1(k7_finseq_1(A,B)) & v1_finseq_1(k7_finseq_1(A,B)) ) ) ), file(finseq_1,k7_finseq_1), []). fof(dt_k8_finseq_1,axiom,( ! [A,B,C] : ( ( m1_finseq_1(B,A) & m1_finseq_1(C,A) ) => m2_finseq_1(k8_finseq_1(A,B,C),A) ) ), file(finseq_1,k8_finseq_1), []). fof(dt_l1_metric_1,axiom,( ! [A] : ( l1_metric_1(A) => l1_struct_0(A) ) ), file(metric_1,l1_metric_1), []). fof(dt_l1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => l1_struct_0(A) ) ), file(pre_topc,l1_pre_topc), []). fof(dt_l1_struct_0,axiom,( $true ), file(struct_0,l1_struct_0), []). fof(dt_m1_finseq_1,axiom,( ! [A,B] : ( m1_finseq_1(B,A) => ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) ) ), file(finseq_1,m1_finseq_1), []). fof(dt_m1_finseq_2,axiom,( $true ), file(finseq_2,m1_finseq_2), []). fof(dt_m1_matrix_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,k5_numbers) & m1_subset_1(C,k5_numbers) ) => ! [D] : ( m1_matrix_1(D,A,B,C) => ( v1_matrix_1(D) & m2_finseq_1(D,k3_finseq_2(A)) ) ) ) ), file(matrix_1,m1_matrix_1), []). fof(dt_m1_relset_1,axiom,( $true ), file(relset_1,m1_relset_1), []). fof(dt_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), []). fof(dt_m2_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_metric_1,axiom,( ! [A] : ( l1_metric_1(A) => ( v1_funct_1(u1_metric_1(A)) & v1_funct_2(u1_metric_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) & m2_relset_1(u1_metric_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) ) ) ), file(metric_1,u1_metric_1), []). fof(dt_u1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => m1_subset_1(u1_pre_topc(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ), file(pre_topc,u1_pre_topc), []). fof(dt_u1_struct_0,axiom,( $true ), file(struct_0,u1_struct_0), []). fof(existence_l1_metric_1,axiom,( ? [A] : l1_metric_1(A) ), file(metric_1,l1_metric_1), []). fof(existence_l1_pre_topc,axiom,( ? [A] : l1_pre_topc(A) ), file(pre_topc,l1_pre_topc), []). fof(existence_l1_struct_0,axiom,( ? [A] : l1_struct_0(A) ), file(struct_0,l1_struct_0), []). fof(existence_m1_finseq_1,axiom,( ! [A] : ? [B] : m1_finseq_1(B,A) ), file(finseq_1,m1_finseq_1), []). fof(existence_m1_finseq_2,axiom,( ! [A] : ? [B] : m1_finseq_2(B,A) ), file(finseq_2,m1_finseq_2), []). fof(existence_m1_matrix_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,k5_numbers) & m1_subset_1(C,k5_numbers) ) => ? [D] : m1_matrix_1(D,A,B,C) ) ), file(matrix_1,m1_matrix_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_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(fc10_membered,axiom,( ! [A] : ( v1_int_1(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) & v3_membered(k1_tarski(A)) & v4_membered(k1_tarski(A)) ) ) ), file(membered,fc10_membered), []). fof(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_membered,axiom,( ! [A] : ( v4_ordinal2(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) & v3_membered(k1_tarski(A)) & v4_membered(k1_tarski(A)) & v5_membered(k1_tarski(A)) ) ) ), file(membered,fc11_membered), []). fof(fc11_sprect_1,axiom,( ! [A] : ( ( v1_sprect_1(A) & m1_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ( ~ v1_xboole_0(k5_topreal1(2,A)) & v2_jordan1(k5_topreal1(2,A)) & v6_compts_1(k5_topreal1(2,A),k15_euclid(2)) ) ) ), file(sprect_1,fc11_sprect_1), []). fof(fc11_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v2_xreal_0(k2_xcmplx_0(A,B)) & v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc11_xreal_0), []). fof(fc12_membered,axiom,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => v1_membered(k2_tarski(A,B)) ) ), file(membered,fc12_membered), []). fof(fc12_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & ~ v2_xreal_0(k2_xcmplx_0(B,A)) & v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc12_xreal_0), []). fof(fc13_membered,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) ) ) ), file(membered,fc13_membered), []). fof(fc14_membered,axiom,( ! [A,B] : ( ( v1_rat_1(A) & v1_rat_1(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) & v3_membered(k2_tarski(A,B)) ) ) ), file(membered,fc14_membered), []). fof(fc15_membered,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) & v3_membered(k2_tarski(A,B)) & v4_membered(k2_tarski(A,B)) ) ) ), file(membered,fc15_membered), []). fof(fc16_membered,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) & v3_membered(k2_tarski(A,B)) & v4_membered(k2_tarski(A,B)) & v5_membered(k2_tarski(A,B)) ) ) ), file(membered,fc16_membered), []). fof(fc17_membered,axiom,( ! [A,B,C] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) & v1_xcmplx_0(C) ) => v1_membered(k1_enumset1(A,B,C)) ) ), file(membered,fc17_membered), []). fof(fc18_membered,axiom,( ! [A,B,C] : ( ( v1_xreal_0(A) & v1_xreal_0(B) & v1_xreal_0(C) ) => ( v1_membered(k1_enumset1(A,B,C)) & v2_membered(k1_enumset1(A,B,C)) ) ) ), file(membered,fc18_membered), []). fof(fc19_membered,axiom,( ! [A,B,C] : ( ( v1_rat_1(A) & v1_rat_1(B) & v1_rat_1(C) ) => ( v1_membered(k1_enumset1(A,B,C)) & v2_membered(k1_enumset1(A,B,C)) & v3_membered(k1_enumset1(A,B,C)) ) ) ), file(membered,fc19_membered), []). fof(fc1_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( ~ v3_struct_0(k14_euclid(A)) & v1_metric_1(k14_euclid(A)) & v6_metric_1(k14_euclid(A)) & v7_metric_1(k14_euclid(A)) & v8_metric_1(k14_euclid(A)) & v9_metric_1(k14_euclid(A)) ) ) ), file(euclid,fc1_euclid), []). fof(fc1_goboard2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & m1_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ( ~ v1_xboole_0(k2_goboard1(A)) & v1_relat_1(k2_goboard1(A)) & v1_funct_1(k2_goboard1(A)) & v1_finset_1(k2_goboard1(A)) & v1_finseq_1(k2_goboard1(A)) ) ) ), file(goboard2,fc1_goboard2), []). fof(fc1_sprect_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v5_seqm_3(A) ) => v1_realset1(k2_relat_1(A)) ) ), file(sprect_1,fc1_sprect_1), []). fof(fc1_sprect_3,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & ~ v5_seqm_3(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & v2_goboard5(A) & m1_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ( ~ v1_xboole_0(k5_topreal1(2,A)) & ~ v1_sppol_1(k5_topreal1(2,A)) & ~ v2_sppol_1(k5_topreal1(2,A)) & v6_compts_1(k5_topreal1(2,A),k15_euclid(2)) & v1_topreal2(k5_topreal1(2,A)) ) ) ), file(sprect_3,fc1_sprect_3), []). 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(fc20_membered,axiom,( ! [A,B,C] : ( ( v1_int_1(A) & v1_int_1(B) & v1_int_1(C) ) => ( v1_membered(k1_enumset1(A,B,C)) & v2_membered(k1_enumset1(A,B,C)) & v3_membered(k1_enumset1(A,B,C)) & v4_membered(k1_enumset1(A,B,C)) ) ) ), file(membered,fc20_membered), []). fof(fc21_membered,axiom,( ! [A,B,C] : ( ( v4_ordinal2(A) & v4_ordinal2(B) & v4_ordinal2(C) ) => ( v1_membered(k1_enumset1(A,B,C)) & v2_membered(k1_enumset1(A,B,C)) & v3_membered(k1_enumset1(A,B,C)) & v4_membered(k1_enumset1(A,B,C)) & v5_membered(k1_enumset1(A,B,C)) ) ) ), file(membered,fc21_membered), []). fof(fc22_membered,axiom,( ! [A,B] : ( ( v1_membered(A) & v1_membered(B) ) => v1_membered(k2_xboole_0(A,B)) ) ), file(membered,fc22_membered), []). fof(fc23_membered,axiom,( ! [A,B] : ( ( v2_membered(A) & v2_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc23_membered), []). fof(fc24_membered,axiom,( ! [A,B] : ( ( v3_membered(A) & v3_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) & v3_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc24_membered), []). fof(fc25_membered,axiom,( ! [A,B] : ( ( v4_membered(A) & v4_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) & v3_membered(k2_xboole_0(A,B)) & v4_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc25_membered), []). fof(fc26_membered,axiom,( ! [A,B] : ( ( v5_membered(A) & v5_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) & v3_membered(k2_xboole_0(A,B)) & v4_membered(k2_xboole_0(A,B)) & v5_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc26_membered), []). fof(fc2_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( ~ v3_struct_0(k15_euclid(A)) & v1_pre_topc(k15_euclid(A)) & v2_pre_topc(k15_euclid(A)) ) ) ), file(euclid,fc2_euclid), []). 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_goboard2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & m1_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ( ~ v1_xboole_0(k3_goboard1(A)) & v1_relat_1(k3_goboard1(A)) & v1_funct_1(k3_goboard1(A)) & v1_finset_1(k3_goboard1(A)) & v1_finseq_1(k3_goboard1(A)) ) ) ), file(goboard2,fc2_goboard2), []). fof(fc2_membered,axiom, ( ~ v1_xboole_0(k1_numbers) & v1_membered(k1_numbers) & v2_membered(k1_numbers) ), file(membered,fc2_membered), []). fof(fc2_sprect_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & ~ v1_realset1(B) & m1_finseq_1(B,u1_struct_0(k15_euclid(A))) ) => ~ v1_xboole_0(k5_topreal1(A,B)) ) ), file(sprect_1,fc2_sprect_1), []). fof(fc3_goboard2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_goboard1(A) & m1_finseq_1(A,k1_numbers) & ~ v1_xboole_0(B) & v1_goboard1(B) & m1_finseq_1(B,k1_numbers) ) => ( v1_relat_1(k1_goboard2(A,B)) & ~ v3_relat_1(k1_goboard2(A,B)) & v1_funct_1(k1_goboard2(A,B)) & v1_finset_1(k1_goboard2(A,B)) & v1_finseq_1(k1_goboard2(A,B)) & v1_matrix_1(k1_goboard2(A,B)) & v3_goboard1(k1_goboard2(A,B)) & v4_goboard1(k1_goboard2(A,B)) & v5_goboard1(k1_goboard2(A,B)) & v6_goboard1(k1_goboard2(A,B)) ) ) ), file(goboard2,fc3_goboard2), []). fof(fc3_sprect_1,axiom,( ! [A] : ( m1_finseq_1(A,u1_struct_0(k15_euclid(2))) => v6_compts_1(k5_topreal1(2,A),k15_euclid(2)) ) ), file(sprect_1,fc3_sprect_1), []). fof(fc3_sprect_2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & v2_goboard5(A) & ~ v5_seqm_3(A) & m1_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ( ~ v1_xboole_0(k5_topreal1(2,A)) & v1_topreal2(k5_topreal1(2,A)) & v6_compts_1(k5_topreal1(2,A),k15_euclid(2)) ) ) ), file(sprect_2,fc3_sprect_2), []). 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_goboard2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & m1_finseq_1(A,k1_numbers) ) => ( ~ v1_xboole_0(k2_goboard2(A)) & v1_relat_1(k2_goboard2(A)) & v1_funct_1(k2_goboard2(A)) & v1_finset_1(k2_goboard2(A)) & v1_finseq_1(k2_goboard2(A)) & v1_goboard1(k2_goboard2(A)) ) ) ), file(goboard2,fc4_goboard2), []). fof(fc5_goboard2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & m1_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ( v1_relat_1(k3_goboard2(A)) & ~ v3_relat_1(k3_goboard2(A)) & v1_funct_1(k3_goboard2(A)) & v1_finset_1(k3_goboard2(A)) & v1_finseq_1(k3_goboard2(A)) & v1_matrix_1(k3_goboard2(A)) & v3_goboard1(k3_goboard2(A)) & v4_goboard1(k3_goboard2(A)) & v5_goboard1(k3_goboard2(A)) & v6_goboard1(k3_goboard2(A)) ) ) ), file(goboard2,fc5_goboard2), []). fof(fc5_membered,axiom, ( v1_membered(k5_ordinal2) & v2_membered(k5_ordinal2) & v3_membered(k5_ordinal2) & v4_membered(k5_ordinal2) & v5_membered(k5_ordinal2) ), file(membered,fc5_membered), []). fof(fc6_membered,axiom, ( v1_xboole_0(k1_xboole_0) & v1_membered(k1_xboole_0) & v2_membered(k1_xboole_0) & v3_membered(k1_xboole_0) & v4_membered(k1_xboole_0) & v5_membered(k1_xboole_0) ), file(membered,fc6_membered), []). fof(fc7_membered,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_membered(k1_tarski(A)) ) ), file(membered,fc7_membered), []). 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_membered,axiom,( ! [A] : ( v1_xreal_0(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) ) ) ), file(membered,fc8_membered), []). 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_membered,axiom,( ! [A] : ( v1_rat_1(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) & v3_membered(k1_tarski(A)) ) ) ), file(membered,fc9_membered), []). fof(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_metric_1,axiom,( ! [A,B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers) & m1_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) ) => ! [C,D] : ( g1_metric_1(A,B) = g1_metric_1(C,D) => ( A = C & B = D ) ) ) ), file(metric_1,g1_metric_1), []). fof(free_g1_pre_topc,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) => ! [C,D] : ( g1_pre_topc(A,B) = g1_pre_topc(C,D) => ( A = C & B = D ) ) ) ), file(pre_topc,g1_pre_topc), []). fof(idempotence_k2_xboole_0,axiom,( ! [A,B] : k2_xboole_0(A,A) = A ), file(xboole_0,k2_xboole_0), []). fof(rc1_arytm_3,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ), file(arytm_3,rc1_arytm_3), []). fof(rc1_membered,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ), file(membered,rc1_membered), []). fof(rc1_sprect_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) & v5_seqm_3(A) ) ), file(sprect_1,rc1_sprect_1), []). fof(rc1_sprect_3,axiom,( ? [A] : ( m1_finseq_1(A,u1_struct_0(k15_euclid(2))) & ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & ~ v1_realset1(A) & v1_sprect_1(A) & ~ v5_seqm_3(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & v2_goboard5(A) ) ), file(sprect_3,rc1_sprect_3), []). fof(rc1_xreal_0,axiom,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(xreal_0,rc1_xreal_0), []). fof(rc2_sprect_1,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) & ~ v1_xboole_0(A) & ~ v1_sppol_1(A) & ~ v2_sppol_1(A) & v6_compts_1(A,k15_euclid(2)) ) ), file(sprect_1,rc2_sprect_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_struct_0,axiom,( ? [A] : ( l1_struct_0(A) & ~ v3_struct_0(A) ) ), file(struct_0,rc3_struct_0), []). 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_sprect_1,axiom,( ? [A] : ( m1_finseq_1(A,u1_struct_0(k15_euclid(2))) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) & v1_sprect_1(A) ) ), file(sprect_1,rc4_sprect_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_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_k1_goboard1,axiom,( ! [A,B] : ( ( m1_finseq_1(A,k1_numbers) & m1_subset_1(B,k5_numbers) ) => k1_goboard1(A,B) = k1_funct_1(A,B) ) ), file(goboard1,k1_goboard1), []). fof(redefinition_k1_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k1_nat_1(A,B) = k2_xcmplx_0(A,B) ) ), file(nat_1,k1_nat_1), []). fof(redefinition_k2_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => k2_finseq_1(A) = k1_finseq_1(A) ) ), file(finseq_1,k2_finseq_1), []). fof(redefinition_k2_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_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => k3_finseq_1(A) = k1_card_1(A) ) ), file(finseq_1,k3_finseq_1), []). fof(redefinition_k3_finseq_2,axiom,( ! [A] : k3_finseq_2(A) = k13_finseq_1(A) ), file(finseq_2,k3_finseq_2), []). fof(redefinition_k3_finseq_4,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) & m1_subset_1(C,A) & m1_subset_1(D,A) ) => k3_finseq_4(A,B,C,D) = k11_finseq_1(B,C,D) ) ), file(finseq_4,k3_finseq_4), []). fof(redefinition_k4_card_1,axiom,( ! [A] : ( v1_finset_1(A) => k4_card_1(A) = k1_card_1(A) ) ), file(card_1,k4_card_1), []). fof(redefinition_k4_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => k4_finseq_1(A) = k1_relat_1(A) ) ), file(finseq_1,k4_finseq_1), []). fof(redefinition_k5_numbers,axiom,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), []). fof(redefinition_k6_matrix_2,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) & m1_subset_1(C,A) & m1_subset_1(D,A) & m1_subset_1(E,A) ) => k6_matrix_2(A,B,C,D,E) = k3_matrix_2(B,C,D,E) ) ), file(matrix_2,k6_matrix_2), []). fof(redefinition_k8_finseq_1,axiom,( ! [A,B,C] : ( ( m1_finseq_1(B,A) & m1_finseq_1(C,A) ) => k8_finseq_1(A,B,C) = k7_finseq_1(B,C) ) ), file(finseq_1,k8_finseq_1), []). fof(redefinition_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(t105_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( k21_euclid(k25_pscomp_1(A)) = k21_euclid(k35_pscomp_1(A)) & k21_euclid(k25_pscomp_1(A)) = k21_euclid(k34_pscomp_1(A)) & k21_euclid(k35_pscomp_1(A)) = k21_euclid(k34_pscomp_1(A)) & k21_euclid(k35_pscomp_1(A)) = k21_euclid(k24_pscomp_1(A)) & k21_euclid(k34_pscomp_1(A)) = k21_euclid(k24_pscomp_1(A)) ) ) ), file(pscomp_1,t105_pscomp_1), []). fof(t115_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( k22_euclid(k22_pscomp_1(A)) = k22_euclid(k37_pscomp_1(A)) & k22_euclid(k22_pscomp_1(A)) = k22_euclid(k36_pscomp_1(A)) & k22_euclid(k37_pscomp_1(A)) = k22_euclid(k36_pscomp_1(A)) & k22_euclid(k37_pscomp_1(A)) = k22_euclid(k25_pscomp_1(A)) & k22_euclid(k36_pscomp_1(A)) = k22_euclid(k25_pscomp_1(A)) ) ) ), file(pscomp_1,t115_pscomp_1), []). fof(t12_sprect_3,axiom,( ! [A,B,C,D] : k2_matrix_1(k3_matrix_2(A,B,C,D)) = k2_enumset1(k4_tarski(1,1),k4_tarski(1,2),k4_tarski(2,1),k4_tarski(2,2)) ), file(sprect_3,t12_sprect_3), []). fof(t12_xboole_1,axiom,( ! [A,B] : ( r1_tarski(A,B) => k2_xboole_0(A,B) = B ) ), file(xboole_1,t12_xboole_1), []). fof(t147_finseq_2,axiom,( ! [A,B] : k2_relat_1(k10_finseq_1(A,B)) = k2_tarski(A,B) ), file(finseq_2,t147_finseq_2), []). fof(t148_finseq_2,axiom,( ! [A,B,C] : k2_relat_1(k11_finseq_1(A,B,C)) = k1_enumset1(A,B,C) ), file(finseq_2,t148_finseq_2), []). fof(t1_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k2_xcmplx_0(A,0) = A ) ), file(arithm,t1_arithm), []). fof(t1_boole,axiom,( ! [A] : k2_xboole_0(A,k1_xboole_0) = A ), file(boole,t1_boole), []). fof(t1_numerals,axiom,( m1_subset_1(0,k5_numbers) ), file(numerals,t1_numerals), []). fof(t1_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & v2_xreal_0(A) ) => v2_xreal_0(B) ) ) ) ), file(real,t1_real), []). fof(t1_subset,axiom,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), []). fof(t27_finseq_3,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( v4_ordinal2(B) => ( r2_hidden(B,k4_finseq_1(A)) <=> ( r1_xreal_0(1,B) & r1_xreal_0(B,k3_finseq_1(A)) ) ) ) ) ), file(finseq_3,t27_finseq_3), []). fof(t29_finseq_3,axiom,( ! [A,B] : k4_finseq_1(k10_finseq_1(A,B)) = k2_finseq_1(2) ), file(finseq_3,t29_finseq_3), []). 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(t33_zfmisc_1,axiom,( ! [A,B,C,D] : ( k4_tarski(A,B) = k4_tarski(C,D) => ( A = C & B = D ) ) ), file(zfmisc_1,t33_zfmisc_1), []). fof(t35_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => k3_finseq_1(k7_finseq_1(A,B)) = k1_nat_1(k3_finseq_1(A),k3_finseq_1(B)) ) ) ), file(finseq_1,t35_finseq_1), []). fof(t3_matrix_2,axiom,( ! [A,B,C,D] : ( k3_finseq_1(k3_matrix_2(A,B,C,D)) = 2 & k1_matrix_1(k3_matrix_2(A,B,C,D)) = 2 & k2_matrix_1(k3_matrix_2(A,B,C,D)) = k2_zfmisc_1(k2_finseq_1(2),k2_finseq_1(2)) ) ), file(matrix_2,t3_matrix_2), []). 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(t44_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => k2_relat_1(k7_finseq_1(A,B)) = k2_xboole_0(k2_relat_1(A),k2_relat_1(B)) ) ) ), file(finseq_1,t44_finseq_1), []). fof(t4_finseq_1,axiom, ( k2_finseq_1(0) = k1_xboole_0 & k2_finseq_1(1) = k1_tarski(1) & k2_finseq_1(2) = k2_tarski(1,2) ), file(finseq_1,t4_finseq_1), []). 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(t55_sprect_2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & v2_goboard5(A) & ~ v5_seqm_3(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ~ r1_xreal_0(k21_euclid(k33_pscomp_1(k5_topreal1(2,A))),k21_euclid(k32_pscomp_1(k5_topreal1(2,A)))) ) ), file(sprect_2,t55_sprect_2), []). fof(t57_euclid,axiom,( ! [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) => A = k23_euclid(k21_euclid(A),k22_euclid(A)) ) ), file(euclid,t57_euclid), []). 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(t61_finseq_1,axiom,( ! [A,B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v1_finseq_1(C) ) => ( C = k10_finseq_1(A,B) <=> ( k3_finseq_1(C) = 2 & k1_funct_1(C,1) = A & k1_funct_1(C,2) = B ) ) ) ), file(finseq_1,t61_finseq_1), []). fof(t61_sprect_2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & v2_goboard5(A) & ~ v5_seqm_3(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ~ r1_xreal_0(k22_euclid(k31_pscomp_1(k5_topreal1(2,A))),k22_euclid(k30_pscomp_1(k5_topreal1(2,A)))) ) ), file(sprect_2,t61_sprect_2), []). fof(t62_finseq_1,axiom,( ! [A,B,C,D] : ( ( v1_relat_1(D) & v1_funct_1(D) & v1_finseq_1(D) ) => ( D = k11_finseq_1(A,B,C) <=> ( k3_finseq_1(D) = 3 & k1_funct_1(D,1) = A & k1_funct_1(D,2) = B & k1_funct_1(D,3) = C ) ) ) ), file(finseq_1,t62_finseq_1), []). fof(t6_boole,axiom,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), []). fof(t6_cqc_the1,axiom,( ! [A] : ( v4_ordinal2(A) => ~ ( r1_xreal_0(A,5) & A != 0 & A != 1 & A != 2 & A != 3 & A != 4 & A != 5 ) ) ), file(cqc_the1,t6_cqc_the1), []). fof(t6_matrix_2,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m1_subset_1(B,A) => ! [C] : ( m1_subset_1(C,A) => ! [D] : ( m1_subset_1(D,A) => ! [E] : ( m1_subset_1(E,A) => ( k3_matrix_1(A,k6_matrix_2(A,B,C,D,E),1,1) = B & k3_matrix_1(A,k6_matrix_2(A,B,C,D,E),1,2) = C & k3_matrix_1(A,k6_matrix_2(A,B,C,D,E),2,1) = D & k3_matrix_1(A,k6_matrix_2(A,B,C,D,E),2,2) = E ) ) ) ) ) ) ), file(matrix_2,t6_matrix_2), []). 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(t76_card_2,axiom,( ! [A,B] : ( A != B => k4_card_1(k2_tarski(A,B)) = 2 ) ), file(card_2,t76_card_2), []). 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(t85_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( k21_euclid(k22_pscomp_1(A)) = k21_euclid(k30_pscomp_1(A)) & k21_euclid(k22_pscomp_1(A)) = k21_euclid(k31_pscomp_1(A)) & k21_euclid(k30_pscomp_1(A)) = k21_euclid(k31_pscomp_1(A)) & k21_euclid(k30_pscomp_1(A)) = k21_euclid(k23_pscomp_1(A)) & k21_euclid(k31_pscomp_1(A)) = k21_euclid(k23_pscomp_1(A)) ) ) ), file(pscomp_1,t85_pscomp_1), []). fof(t86_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,C) & r1_xreal_0(C,k3_finseq_1(B)) ) => k1_funct_1(k7_finseq_1(A,B),k1_nat_1(k3_finseq_1(A),C)) = k1_funct_1(B,C) ) ) ) ) ), file(finseq_1,t86_finseq_1), []). fof(t8_boole,axiom,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), []). fof(t8_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,B) & ~ v3_xreal_0(B) & ~ v2_xreal_0(A) ) ) ) ), file(real,t8_real), []). fof(t90_sprect_1,axiom,( ! [A] : ( ( v1_sprect_1(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => k3_finseq_1(A) = 5 ) ), file(sprect_1,t90_sprect_1), []). fof(t91_sprect_1,axiom,( ! [A] : ( ( v1_sprect_1(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ( k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,1) = k32_pscomp_1(k5_topreal1(2,A)) & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,1) = k31_pscomp_1(k5_topreal1(2,A)) ) ) ), file(sprect_1,t91_sprect_1), []). fof(t92_sprect_1,axiom,( ! [A] : ( ( v1_sprect_1(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ( k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,2) = k33_pscomp_1(k5_topreal1(2,A)) & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,2) = k34_pscomp_1(k5_topreal1(2,A)) ) ) ), file(sprect_1,t92_sprect_1), []). fof(t93_sprect_1,axiom,( ! [A] : ( ( v1_sprect_1(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ( k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,3) = k36_pscomp_1(k5_topreal1(2,A)) & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,3) = k35_pscomp_1(k5_topreal1(2,A)) ) ) ), file(sprect_1,t93_sprect_1), []). fof(t94_sprect_1,axiom,( ! [A] : ( ( v1_sprect_1(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ( k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,4) = k37_pscomp_1(k5_topreal1(2,A)) & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,4) = k30_pscomp_1(k5_topreal1(2,A)) ) ) ), file(sprect_1,t94_sprect_1), []). fof(t95_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( k22_euclid(k23_pscomp_1(A)) = k22_euclid(k32_pscomp_1(A)) & k22_euclid(k23_pscomp_1(A)) = k22_euclid(k33_pscomp_1(A)) & k22_euclid(k32_pscomp_1(A)) = k22_euclid(k33_pscomp_1(A)) & k22_euclid(k32_pscomp_1(A)) = k22_euclid(k24_pscomp_1(A)) & k22_euclid(k33_pscomp_1(A)) = k22_euclid(k24_pscomp_1(A)) ) ) ), file(pscomp_1,t95_pscomp_1), []).