% Mizar problem: t23_pencil_3,pencil_3,1551,52 fof(t23_pencil_3,conjecture,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_finset_1(A) ) => ! [B] : ( ( v15_pencil_1(B) & m1_pboole(B,A) ) => ( ! [C] : ( m1_subset_1(C,A) => v9_pencil_1(k2_pencil_1(A,B,C)) ) => ! [C] : ( m1_pencil_2(C,A,B) => ! [D] : ( m1_pencil_2(D,A,B) => ( r1_xboole_0(C,D) => ! [E] : ( ( ~ v13_pencil_1(E) & v16_pencil_1(E,A) & m4_pboole(E,A,k12_pralg_1(A,B)) ) => ! [F] : ( ( ~ v13_pencil_1(F) & v16_pencil_1(F,A) & m4_pboole(F,A,k12_pralg_1(A,B)) ) => ( ( C = k4_card_3(E) & D = k4_card_3(F) ) => ( k3_pencil_1(A,E) = k3_pencil_1(A,F) <=> ? [G] : ( m2_finseq_1(G,k1_zfmisc_1(u1_struct_0(k6_pencil_1(A,B)))) & k1_funct_1(G,1) = C & k1_funct_1(G,k3_finseq_1(G)) = D & ! [H] : ( m2_subset_1(H,k1_numbers,k5_numbers) => ( r2_hidden(H,k4_finseq_1(G)) => m1_pencil_2(k1_funct_1(G,H),A,B) ) ) & ! [H] : ( m2_subset_1(H,k1_numbers,k5_numbers) => ( r1_xreal_0(1,H) => ( r1_xreal_0(k3_finseq_1(G),H) | ! [I] : ( m1_pencil_2(I,A,B) => ! [J] : ( m1_pencil_2(J,A,B) => ( ( I = k1_funct_1(G,H) & J = k1_funct_1(G,k1_nat_1(H,1)) ) => ( r1_xboole_0(I,J) & r1_pencil_3(A,B,I,J) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ), inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc1_arytm_3,cc1_finset_1,cc1_pboole,cc1_pencil_1,cc1_realset1,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,cc2_finset_1,cc2_pboole,cc2_pencil_1,cc2_xreal_0,cc3_arytm_3,cc3_pencil_1,cc3_xreal_0,cc4_pencil_1,cc4_xreal_0,cc5_pencil_1,cc5_xreal_0,cc6_pencil_1,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,commutativity_k1_nat_1,commutativity_k2_tarski,commutativity_k2_xboole_0,commutativity_k2_xcmplx_0,commutativity_k4_subset_1,connectedness_r1_xreal_0,d1_pencil_3,d1_tarski,d1_xboole_0,d21_pencil_1,d23_pboole,d2_pencil_2,d2_xboole_0,d3_pboole,d3_pre_topc,d4_funct_1,d4_realset1,d5_funct_1,d5_pboole,d5_realset2,d5_tarski,dt_k10_pralg_1,dt_k12_pralg_1,dt_k1_card_1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_ordinal1,dt_k1_pencil_1,dt_k1_pencil_3,dt_k1_relat_1,dt_k1_struct_0,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_funct_7,dt_k2_pencil_1,dt_k2_polynom1,dt_k2_pre_topc,dt_k2_relat_1,dt_k2_tarski,dt_k2_xboole_0,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k3_finseq_1,dt_k3_graph_2,dt_k3_pencil_1,dt_k4_card_3,dt_k4_finseq_1,dt_k4_graph_2,dt_k4_ordinal2,dt_k4_subset_1,dt_k4_tarski,dt_k5_numbers,dt_k5_ordinal2,dt_k5_pencil_1,dt_k5_relset_1,dt_k6_pencil_1,dt_k6_xcmplx_0,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_finseq_1,dt_m1_pboole,dt_m1_pencil_2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_relset_1,dt_m2_subset_1,dt_m4_pboole,dt_u1_struct_0,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_finseq_1,existence_m1_pboole,existence_m1_pencil_2,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_relset_1,existence_m2_subset_1,existence_m4_pboole,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_finset_1,fc15_xreal_0,fc16_xreal_0,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc1_arytm_3,fc1_finset_1,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc2_arytm_3,fc2_finset_1,fc2_pboole,fc2_subset_1,fc2_xboole_0,fc3_realset1,fc3_subset_1,fc3_xboole_0,fc3_xreal_0,fc4_pencil_1,fc4_subset_1,fc5_xreal_0,fc7_xreal_0,fc8_xreal_0,fc9_finset_1,fc9_xreal_0,fraenkel_a_3_0_pencil_3,fraenkel_a_4_1_pencil_3,idempotence_k2_xboole_0,idempotence_k4_subset_1,irreflexivity_r1_subset_1,irreflexivity_r2_subset_1,rc1_arytm_3,rc1_finset_1,rc1_pboole,rc1_realset1,rc1_subset_1,rc1_xboole_0,rc1_xreal_0,rc2_pboole,rc2_realset1,rc2_subset_1,rc2_xboole_0,rc2_xreal_0,rc3_finset_1,rc3_pboole,rc3_pencil_1,rc3_struct_0,rc3_xreal_0,rc4_finset_1,rc4_pencil_1,rc4_xreal_0,rc5_pboole,rc5_pencil_1,rc5_struct_0,rc6_pencil_1,rc7_pencil_1,rc8_pencil_1,redefinition_k10_pralg_1,redefinition_k1_nat_1,redefinition_k1_pencil_1,redefinition_k1_struct_0,redefinition_k2_pencil_1,redefinition_k2_polynom1,redefinition_k3_finseq_1,redefinition_k4_finseq_1,redefinition_k4_graph_2,redefinition_k4_subset_1,redefinition_k5_numbers,redefinition_k5_relset_1,redefinition_k6_pencil_1,redefinition_m2_finseq_1,redefinition_m2_relset_1,redefinition_m2_subset_1,redefinition_r1_subset_1,redefinition_r2_subset_1,redefinition_r6_pboole,reflexivity_r1_tarski,reflexivity_r1_xreal_0,reflexivity_r2_pboole,reflexivity_r6_pboole,s1_nat_1__e4_25_1__pencil_3,s1_nat_1__e9_25__pencil_3,spc0_boole,spc1_boole,spc1_numerals,spc2_boole,spc2_numerals,symmetry_r1_subset_1,symmetry_r1_xboole_0,symmetry_r2_subset_1,symmetry_r6_pboole,t10_pencil_3,t11_pencil_3,t12_funct_1,t12_pencil_1,t13_graph_2,t13_pencil_3,t14_graph_2,t14_pencil_3,t15_graph_2,t16_graph_2,t17_graph_2,t18_nat_1,t19_card_2,t19_pencil_3,t1_arithm,t1_boole,t1_numerals,t1_real,t1_subset,t1_xreal_1,t21_nat_1,t21_pencil_3,t22_pencil_3,t25_finseq_1,t26_finseq_3,t27_finseq_3,t28_nat_1,t29_nat_1,t2_pencil_3,t2_pua2mss1,t2_real,t2_subset,t2_tarski,t2_xreal_1,t33_funct_7,t34_funct_7,t37_zfmisc_1,t38_nat_1,t3_real,t3_subset,t4_arithm,t4_real,t4_subset,t59_card_2,t5_real,t5_subset,t6_boole,t6_real,t7_boole,t7_pencil_3,t7_real,t8_boole,t8_real,t8_xreal_1,t9_funct_1]), [file(pencil_3,t23_pencil_3)]). fof(antisymmetry_r2_hidden,axiom,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), []). 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_finset_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_finset_1(A) ) ), file(finset_1,cc1_finset_1), []). fof(cc1_pboole,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m1_pboole(B,A) => ( v2_relat_1(B) => ~ v3_relat_1(B) ) ) ) ), file(pboole,cc1_pboole), []). fof(cc1_pencil_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v11_pencil_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v2_pralg_1(A) ) ) ), file(pencil_1,cc1_pencil_1), []). fof(cc1_realset1,axiom,( ! [A] : ( ~ v1_realset1(A) => ~ v1_xboole_0(A) ) ), file(realset1,cc1_realset1), []). 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_xreal_0,axiom,( ! [A] : ( v4_ordinal2(A) => v1_xreal_0(A) ) ), file(xreal_0,cc1_xreal_0), []). 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_finset_1,axiom,( ! [A] : ( v1_finset_1(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_finset_1(B) ) ) ), file(finset_1,cc2_finset_1), []). fof(cc2_pboole,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m1_pboole(B,A) => ( v3_relat_1(B) => ~ v2_relat_1(B) ) ) ) ), file(pboole,cc2_pboole), []). fof(cc2_pencil_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v14_pencil_1(A) ) => ( v1_relat_1(A) & v4_waybel_3(A) ) ) ), file(pencil_1,cc2_pencil_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_pencil_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v15_pencil_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v4_waybel_3(A) & v2_pralg_1(A) & v11_pencil_1(A) ) ) ), file(pencil_1,cc3_pencil_1), []). fof(cc3_xreal_0,axiom,( ! [A] : ( ( v1_xreal_0(A) & v2_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc3_xreal_0), []). fof(cc4_pencil_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v11_pencil_1(A) & v15_pencil_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v2_pralg_1(A) & v11_pencil_1(A) & v12_pencil_1(A) ) ) ), file(pencil_1,cc4_pencil_1), []). fof(cc4_xreal_0,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v2_xreal_0(A) ) ) ), file(xreal_0,cc4_xreal_0), []). fof(cc5_pencil_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v11_pencil_1(A) & v15_pencil_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v4_waybel_3(A) & v2_pralg_1(A) & v11_pencil_1(A) & v14_pencil_1(A) ) ) ), file(pencil_1,cc5_pencil_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_pencil_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m1_pboole(B,A) => ( ( ~ v13_pencil_1(B) & v16_pencil_1(B,A) ) => v2_relat_1(B) ) ) ) ), file(pencil_1,cc6_pencil_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(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(commutativity_k4_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k4_subset_1(A,B,C) = k4_subset_1(A,C,B) ) ), file(subset_1,k4_subset_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(d1_pencil_3,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_finset_1(A) ) => ! [B] : ( m1_pboole(B,A) => ! [C] : ( m1_pboole(C,A) => k1_pencil_3(A,B,C) = k1_card_1(a_3_0_pencil_3(A,B,C)) ) ) ) ), file(pencil_3,d1_pencil_3), []). fof(d1_tarski,axiom,( ! [A,B] : ( B = k1_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> C = A ) ) ), file(tarski,d1_tarski), []). fof(d1_xboole_0,axiom,( ! [A] : ( A = k1_xboole_0 <=> ! [B] : ~ r2_hidden(B,A) ) ), file(xboole_0,d1_xboole_0), []). fof(d21_pencil_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( ~ v13_pencil_1(B) & v16_pencil_1(B,A) & m1_pboole(B,A) ) => ! [C] : ( m1_subset_1(C,A) => ( C = k3_pencil_1(A,B) <=> ~ v1_realset1(k1_funct_1(B,C)) ) ) ) ) ), file(pencil_1,d21_pencil_1), []). fof(d23_pboole,axiom,( ! [A,B] : ( m1_pboole(B,A) => ! [C] : ( m1_pboole(C,A) => ( m4_pboole(C,A,B) <=> r2_pboole(A,C,B) ) ) ) ), file(pboole,d23_pboole), []). fof(d2_pencil_2,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( v11_pencil_1(B) & v14_pencil_1(B) & m1_pboole(B,A) ) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pencil_1(A,B)))) => ( m1_pencil_2(C,A,B) <=> ? [D] : ( ~ v13_pencil_1(D) & v16_pencil_1(D,A) & m4_pboole(D,A,k12_pralg_1(A,B)) & C = k4_card_3(D) & k1_funct_1(D,k3_pencil_1(A,D)) = k2_pre_topc(k1_pencil_1(A,B,k3_pencil_1(A,D))) ) ) ) ) ) ), file(pencil_2,d2_pencil_2), []). fof(d2_xboole_0,axiom,( ! [A,B,C] : ( C = k2_xboole_0(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( r2_hidden(D,A) | r2_hidden(D,B) ) ) ) ), file(xboole_0,d2_xboole_0), []). fof(d3_pboole,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( m1_pboole(B,A) <=> k1_relat_1(B) = A ) ) ), file(pboole,d3_pboole), []). fof(d3_pre_topc,axiom,( ! [A] : ( l1_struct_0(A) => k2_pre_topc(A) = u1_struct_0(A) ) ), file(pre_topc,d3_pre_topc), []). fof(d4_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B,C] : ( ( r2_hidden(B,k1_relat_1(A)) => ( C = k1_funct_1(A,B) <=> r2_hidden(k4_tarski(B,C),A) ) ) & ( ~ r2_hidden(B,k1_relat_1(A)) => ( C = k1_funct_1(A,B) <=> C = k1_xboole_0 ) ) ) ) ), file(funct_1,d4_funct_1), []). fof(d4_realset1,axiom,( ! [A] : ( v1_realset1(A) <=> ~ ( ~ v1_xboole_0(A) & ! [B] : A != k1_tarski(B) ) ) ), file(realset1,d4_realset1), []). fof(d5_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( B = k2_relat_1(A) <=> ! [C] : ( r2_hidden(C,B) <=> ? [D] : ( r2_hidden(D,k1_relat_1(A)) & C = k1_funct_1(A,D) ) ) ) ) ), file(funct_1,d5_funct_1), []). fof(d5_pboole,axiom,( ! [A,B] : ( m1_pboole(B,A) => ! [C] : ( m1_pboole(C,A) => ( r2_pboole(A,B,C) <=> ! [D] : ( r2_hidden(D,A) => r1_tarski(k1_funct_1(B,D),k1_funct_1(C,D)) ) ) ) ) ), file(pboole,d5_pboole), []). fof(d5_realset2,axiom,( ! [A] : ( l1_struct_0(A) => ( v3_realset2(A) <=> v1_realset1(u1_struct_0(A)) ) ) ), file(realset2,d5_realset2), []). fof(d5_tarski,axiom,( ! [A,B] : k4_tarski(A,B) = k2_tarski(k2_tarski(A,B),k1_tarski(A)) ), file(tarski,d5_tarski), []). fof(dt_k10_pralg_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v2_pralg_1(B) & m1_pboole(B,A) & m1_subset_1(C,A) ) => l1_struct_0(k10_pralg_1(A,B,C)) ) ), file(pralg_1,k10_pralg_1), []). fof(dt_k12_pralg_1,axiom,( ! [A,B] : ( ( v2_pralg_1(B) & m1_pboole(B,A) ) => m1_pboole(k12_pralg_1(A,B),A) ) ), file(pralg_1,k12_pralg_1), []). fof(dt_k1_card_1,axiom,( ! [A] : v1_card_1(k1_card_1(A)) ), file(card_1,k1_card_1), []). fof(dt_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), []). fof(dt_k1_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_ordinal1,axiom,( $true ), file(ordinal1,k1_ordinal1), []). fof(dt_k1_pencil_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v11_pencil_1(B) & m1_pboole(B,A) & m1_subset_1(C,A) ) => l1_pre_topc(k1_pencil_1(A,B,C)) ) ), file(pencil_1,k1_pencil_1), []). fof(dt_k1_pencil_3,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_finset_1(A) & m1_pboole(B,A) & m1_pboole(C,A) ) => m2_subset_1(k1_pencil_3(A,B,C),k1_numbers,k5_numbers) ) ), file(pencil_3,k1_pencil_3), []). fof(dt_k1_relat_1,axiom,( $true ), file(relat_1,k1_relat_1), []). fof(dt_k1_struct_0,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & m1_subset_1(B,u1_struct_0(A)) ) => m1_subset_1(k1_struct_0(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ), file(struct_0,k1_struct_0), []). fof(dt_k1_tarski,axiom,( $true ), file(tarski,k1_tarski), []). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), []). fof(dt_k1_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), []). fof(dt_k2_funct_7,axiom,( ! [A,B,C] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( v1_relat_1(k2_funct_7(A,B,C)) & v1_funct_1(k2_funct_7(A,B,C)) ) ) ), file(funct_7,k2_funct_7), []). fof(dt_k2_pencil_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v15_pencil_1(B) & m1_pboole(B,A) & m1_subset_1(C,A) ) => ( ~ v3_struct_0(k2_pencil_1(A,B,C)) & ~ v3_pencil_1(k2_pencil_1(A,B,C)) & ~ v4_pencil_1(k2_pencil_1(A,B,C)) & v5_pencil_1(k2_pencil_1(A,B,C)) & v6_pencil_1(k2_pencil_1(A,B,C)) & l1_pre_topc(k2_pencil_1(A,B,C)) ) ) ), file(pencil_1,k2_pencil_1), []). fof(dt_k2_polynom1,axiom,( ! [A,B,C,D] : ( m1_pboole(B,A) => m1_pboole(k2_polynom1(A,B,C,D),A) ) ), file(polynom1,k2_polynom1), []). fof(dt_k2_pre_topc,axiom,( ! [A] : ( l1_struct_0(A) => m1_subset_1(k2_pre_topc(A),k1_zfmisc_1(u1_struct_0(A))) ) ), file(pre_topc,k2_pre_topc), []). 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_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_graph_2,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(k3_graph_2(A,B)) & v1_funct_1(k3_graph_2(A,B)) & v1_finseq_1(k3_graph_2(A,B)) ) ) ), file(graph_2,k3_graph_2), []). fof(dt_k3_pencil_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v13_pencil_1(B) & v16_pencil_1(B,A) & m1_pboole(B,A) ) => m1_subset_1(k3_pencil_1(A,B),A) ) ), file(pencil_1,k3_pencil_1), []). fof(dt_k4_card_3,axiom,( $true ), file(card_3,k4_card_3), []). 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_graph_2,axiom,( ! [A,B,C] : ( ( m1_finseq_1(B,A) & m1_finseq_1(C,A) ) => m2_finseq_1(k4_graph_2(A,B,C),A) ) ), file(graph_2,k4_graph_2), []). fof(dt_k4_ordinal2,axiom, ( v3_ordinal1(k4_ordinal2) & ~ v1_xboole_0(k4_ordinal2) ), file(ordinal2,k4_ordinal2), []). fof(dt_k4_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => m1_subset_1(k4_subset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(subset_1,k4_subset_1), []). 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_pencil_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v4_waybel_3(B) & v11_pencil_1(B) & m1_pboole(B,A) ) => ( ~ v3_struct_0(k5_pencil_1(A,B)) & l1_pre_topc(k5_pencil_1(A,B)) ) ) ), file(pencil_1,k5_pencil_1), []). fof(dt_k5_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => m1_subset_1(k5_relset_1(A,B,C),k1_zfmisc_1(B)) ) ), file(relset_1,k5_relset_1), []). fof(dt_k6_pencil_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v15_pencil_1(B) & m1_pboole(B,A) ) => ( ~ v3_struct_0(k6_pencil_1(A,B)) & ~ v3_pencil_1(k6_pencil_1(A,B)) & ~ v4_pencil_1(k6_pencil_1(A,B)) & v5_pencil_1(k6_pencil_1(A,B)) & v6_pencil_1(k6_pencil_1(A,B)) & l1_pre_topc(k6_pencil_1(A,B)) ) ) ), file(pencil_1,k6_pencil_1), []). fof(dt_k6_xcmplx_0,axiom,( $true ), file(xcmplx_0,k6_xcmplx_0), []). 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_pboole,axiom,( ! [A,B] : ( m1_pboole(B,A) => ( v1_relat_1(B) & v1_funct_1(B) ) ) ), file(pboole,m1_pboole), []). fof(dt_m1_pencil_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v11_pencil_1(B) & v14_pencil_1(B) & m1_pboole(B,A) ) => ! [C] : ( m1_pencil_2(C,A,B) => m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pencil_1(A,B)))) ) ) ), file(pencil_2,m1_pencil_2), []). fof(dt_m1_relset_1,axiom,( $true ), file(relset_1,m1_relset_1), []). fof(dt_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), []). fof(dt_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_m4_pboole,axiom,( ! [A,B] : ( m1_pboole(B,A) => ! [C] : ( m4_pboole(C,A,B) => m1_pboole(C,A) ) ) ), file(pboole,m4_pboole), []). fof(dt_u1_struct_0,axiom,( $true ), file(struct_0,u1_struct_0), []). 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_pboole,axiom,( ! [A] : ? [B] : m1_pboole(B,A) ), file(pboole,m1_pboole), []). fof(existence_m1_pencil_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v11_pencil_1(B) & v14_pencil_1(B) & m1_pboole(B,A) ) => ? [C] : m1_pencil_2(C,A,B) ) ), file(pencil_2,m1_pencil_2), []). fof(existence_m1_relset_1,axiom,( ! [A,B] : ? [C] : m1_relset_1(C,A,B) ), file(relset_1,m1_relset_1), []). fof(existence_m1_subset_1,axiom,( ! [A] : ? [B] : m1_subset_1(B,A) ), file(subset_1,m1_subset_1), []). fof(existence_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(existence_m4_pboole,axiom,( ! [A,B] : ( m1_pboole(B,A) => ? [C] : m4_pboole(C,A,B) ) ), file(pboole,m4_pboole), []). fof(fc10_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v2_xreal_0(k2_xcmplx_0(B,A)) & ~ v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc10_xreal_0), []). fof(fc11_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v2_xreal_0(k2_xcmplx_0(A,B)) & v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc11_xreal_0), []). fof(fc12_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & ~ v2_xreal_0(k2_xcmplx_0(B,A)) & v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc12_xreal_0), []). fof(fc14_finset_1,axiom,( ! [A,B] : ( ( v1_finset_1(A) & v1_finset_1(B) ) => v1_finset_1(k2_zfmisc_1(A,B)) ) ), file(finset_1,fc14_finset_1), []). fof(fc15_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & ~ v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc15_xreal_0), []). fof(fc16_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & ~ v2_xreal_0(k6_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc16_xreal_0), []). fof(fc17_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(A,B)) & v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v2_xreal_0(k6_xcmplx_0(A,B)) & ~ v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc17_xreal_0), []). fof(fc18_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(B,A)) & v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & ~ v2_xreal_0(k6_xcmplx_0(B,A)) & v3_xreal_0(k6_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc18_xreal_0), []). fof(fc19_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(A,B)) & v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & ~ v2_xreal_0(k6_xcmplx_0(A,B)) & v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc19_xreal_0), []). fof(fc1_arytm_3,axiom, ( ~ v1_xboole_0(k4_ordinal2) & v1_ordinal1(k4_ordinal2) & v2_ordinal1(k4_ordinal2) & v3_ordinal1(k4_ordinal2) & v4_ordinal2(k4_ordinal2) ), file(arytm_3,fc1_arytm_3), []). fof(fc1_finset_1,axiom,( ! [A] : ( ~ v1_xboole_0(k1_tarski(A)) & v1_finset_1(k1_tarski(A)) ) ), file(finset_1,fc1_finset_1), []). fof(fc1_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_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ), file(subset_1,fc1_subset_1), []). fof(fc1_xboole_0,axiom,( v1_xboole_0(k1_xboole_0) ), file(xboole_0,fc1_xboole_0), []). fof(fc20_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(B,A)) & v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & v2_xreal_0(k6_xcmplx_0(B,A)) & ~ v3_xreal_0(k6_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc20_xreal_0), []). fof(fc2_arytm_3,axiom,( ! [A] : ( ( v3_ordinal1(A) & v4_ordinal2(A) ) => ( ~ v1_xboole_0(k1_ordinal1(A)) & v1_ordinal1(k1_ordinal1(A)) & v2_ordinal1(k1_ordinal1(A)) & v3_ordinal1(k1_ordinal1(A)) & v4_ordinal2(k1_ordinal1(A)) ) ) ), file(arytm_3,fc2_arytm_3), []). fof(fc2_finset_1,axiom,( ! [A,B] : ( ~ v1_xboole_0(k2_tarski(A,B)) & v1_finset_1(k2_tarski(A,B)) ) ), file(finset_1,fc2_finset_1), []). fof(fc2_pboole,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v2_relat_1(B) & m1_pboole(B,A) & m1_subset_1(C,A) ) => ~ v1_xboole_0(k1_funct_1(B,C)) ) ), file(pboole,fc2_pboole), []). fof(fc2_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_tarski(A)) ), file(subset_1,fc2_subset_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_realset1,axiom,( ! [A] : ( ~ v1_xboole_0(k1_tarski(A)) & v1_finset_1(k1_tarski(A)) & v1_realset1(k1_tarski(A)) ) ), file(realset1,fc3_realset1), []). fof(fc3_subset_1,axiom,( ! [A,B] : ~ v1_xboole_0(k2_tarski(A,B)) ), file(subset_1,fc3_subset_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(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_pencil_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v13_pencil_1(B) & v16_pencil_1(B,A) & m1_pboole(B,A) ) => ( ~ v1_xboole_0(k4_card_3(B)) & v1_fraenkel(k4_card_3(B)) & ~ v1_realset1(k4_card_3(B)) ) ) ), file(pencil_1,fc4_pencil_1), []). fof(fc4_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ), file(subset_1,fc4_subset_1), []). fof(fc5_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc5_xreal_0), []). fof(fc7_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc7_xreal_0), []). fof(fc8_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v2_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc8_xreal_0), []). fof(fc9_finset_1,axiom,( ! [A,B] : ( ( v1_finset_1(A) & v1_finset_1(B) ) => v1_finset_1(k2_xboole_0(A,B)) ) ), file(finset_1,fc9_finset_1), []). fof(fc9_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v2_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc9_xreal_0), []). fof(fraenkel_a_3_0_pencil_3,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(B) & v1_finset_1(B) & m1_pboole(C,B) & m1_pboole(D,B) ) => ( r2_hidden(A,a_3_0_pencil_3(B,C,D)) <=> ? [E] : ( m1_subset_1(E,B) & A = E & k1_funct_1(C,E) != k1_funct_1(D,E) ) ) ) ), file(pencil_3,a_3_0_pencil_3), []). fof(fraenkel_a_4_1_pencil_3,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(B) & v1_finset_1(B) & v15_pencil_1(C) & m1_pboole(C,B) & ~ v13_pencil_1(D) & v16_pencil_1(D,B) & m4_pboole(D,B,k12_pralg_1(B,C)) & ~ v13_pencil_1(E) & v16_pencil_1(E,B) & m4_pboole(E,B,k12_pralg_1(B,C)) ) => ( r2_hidden(A,a_4_1_pencil_3(B,C,D,E)) <=> ? [F] : ( m1_subset_1(F,B) & A = F & k1_funct_1(D,F) != k1_funct_1(E,F) ) ) ) ), file(pencil_3,a_4_1_pencil_3), []). fof(idempotence_k2_xboole_0,axiom,( ! [A,B] : k2_xboole_0(A,A) = A ), file(xboole_0,k2_xboole_0), []). fof(idempotence_k4_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k4_subset_1(A,B,B) = B ) ), file(subset_1,k4_subset_1), []). fof(irreflexivity_r1_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ~ r1_subset_1(A,A) ) ), file(subset_1,r1_subset_1), []). fof(irreflexivity_r2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ~ r2_subset_1(A,A) ) ), file(subset_1,r2_subset_1), []). 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_finset_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) ) ), file(finset_1,rc1_finset_1), []). fof(rc1_pboole,axiom,( ? [A] : ( v1_relat_1(A) & v3_relat_1(A) & v1_funct_1(A) ) ), file(pboole,rc1_pboole), []). fof(rc1_realset1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_realset1(A) ) ), file(realset1,rc1_realset1), []). fof(rc1_subset_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) ) ) ), file(subset_1,rc1_subset_1), []). fof(rc1_xboole_0,axiom,( ? [A] : v1_xboole_0(A) ), file(xboole_0,rc1_xboole_0), []). fof(rc1_xreal_0,axiom,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(xreal_0,rc1_xreal_0), []). fof(rc2_pboole,axiom,( ! [A] : ? [B] : ( m1_pboole(B,A) & v1_relat_1(B) & v3_relat_1(B) & v1_funct_1(B) ) ), file(pboole,rc2_pboole), []). fof(rc2_realset1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & ~ v1_realset1(A) ) ), file(realset1,rc2_realset1), []). fof(rc2_subset_1,axiom,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) ) ), file(subset_1,rc2_subset_1), []). fof(rc2_xboole_0,axiom,( ? [A] : ~ v1_xboole_0(A) ), file(xboole_0,rc2_xboole_0), []). 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_finset_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & v1_finset_1(B) ) ) ), file(finset_1,rc3_finset_1), []). fof(rc3_pboole,axiom,( ! [A] : ? [B] : ( m1_pboole(B,A) & v1_relat_1(B) & v2_relat_1(B) & v1_funct_1(B) ) ), file(pboole,rc3_pboole), []). fof(rc3_pencil_1,axiom,( ! [A] : ? [B] : ( m1_pboole(B,A) & v1_relat_1(B) & v1_funct_1(B) & v2_pralg_1(B) & v11_pencil_1(B) ) ), file(pencil_1,rc3_pencil_1), []). 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_finset_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & v1_finset_1(B) ) ) ), file(finset_1,rc4_finset_1), []). fof(rc4_pencil_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v2_pralg_1(A) & v11_pencil_1(A) ) ), file(pencil_1,rc4_pencil_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_pboole,axiom,( ! [A,B] : ( ( v2_relat_1(B) & m1_pboole(B,A) ) => ? [C] : ( m4_pboole(C,A,B) & v1_relat_1(C) & v2_relat_1(C) & v1_funct_1(C) ) ) ), file(pboole,rc5_pboole), []). fof(rc5_pencil_1,axiom,( ! [A] : ? [B] : ( m1_pboole(B,A) & v1_relat_1(B) & v1_funct_1(B) & v4_waybel_3(B) & v2_pralg_1(B) & v11_pencil_1(B) & v12_pencil_1(B) & v14_pencil_1(B) & v15_pencil_1(B) ) ), file(pencil_1,rc5_pencil_1), []). 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(rc6_pencil_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v4_waybel_3(B) & v2_pralg_1(B) & m1_pboole(B,A) ) => ? [C] : ( m4_pboole(C,A,k12_pralg_1(A,B)) & v1_relat_1(C) & v2_relat_1(C) & v1_funct_1(C) & v13_pencil_1(C) & v16_pencil_1(C,A) ) ) ), file(pencil_1,rc6_pencil_1), []). fof(rc7_pencil_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v2_pralg_1(B) & v14_pencil_1(B) & m1_pboole(B,A) ) => ? [C] : ( m4_pboole(C,A,k12_pralg_1(A,B)) & v1_relat_1(C) & v2_relat_1(C) & v1_funct_1(C) & ~ v13_pencil_1(C) & v16_pencil_1(C,A) ) ) ), file(pencil_1,rc7_pencil_1), []). fof(rc8_pencil_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_pboole(B,A) & v1_relat_1(B) & v1_funct_1(B) & ~ v13_pencil_1(B) & v16_pencil_1(B,A) ) ) ), file(pencil_1,rc8_pencil_1), []). fof(redefinition_k10_pralg_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v2_pralg_1(B) & m1_pboole(B,A) & m1_subset_1(C,A) ) => k10_pralg_1(A,B,C) = k1_funct_1(B,C) ) ), file(pralg_1,k10_pralg_1), []). fof(redefinition_k1_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k1_nat_1(A,B) = k2_xcmplx_0(A,B) ) ), file(nat_1,k1_nat_1), []). fof(redefinition_k1_pencil_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v11_pencil_1(B) & m1_pboole(B,A) & m1_subset_1(C,A) ) => k1_pencil_1(A,B,C) = k1_funct_1(B,C) ) ), file(pencil_1,k1_pencil_1), []). fof(redefinition_k1_struct_0,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & m1_subset_1(B,u1_struct_0(A)) ) => k1_struct_0(A,B) = k1_tarski(B) ) ), file(struct_0,k1_struct_0), []). fof(redefinition_k2_pencil_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v15_pencil_1(B) & m1_pboole(B,A) & m1_subset_1(C,A) ) => k2_pencil_1(A,B,C) = k1_funct_1(B,C) ) ), file(pencil_1,k2_pencil_1), []). fof(redefinition_k2_polynom1,axiom,( ! [A,B,C,D] : ( m1_pboole(B,A) => k2_polynom1(A,B,C,D) = k2_funct_7(B,C,D) ) ), file(polynom1,k2_polynom1), []). fof(redefinition_k3_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => k3_finseq_1(A) = k1_card_1(A) ) ), file(finseq_1,k3_finseq_1), []). fof(redefinition_k4_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => k4_finseq_1(A) = k1_relat_1(A) ) ), file(finseq_1,k4_finseq_1), []). fof(redefinition_k4_graph_2,axiom,( ! [A,B,C] : ( ( m1_finseq_1(B,A) & m1_finseq_1(C,A) ) => k4_graph_2(A,B,C) = k3_graph_2(B,C) ) ), file(graph_2,k4_graph_2), []). fof(redefinition_k4_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k4_subset_1(A,B,C) = k2_xboole_0(B,C) ) ), file(subset_1,k4_subset_1), []). fof(redefinition_k5_numbers,axiom,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), []). fof(redefinition_k5_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => k5_relset_1(A,B,C) = k2_relat_1(C) ) ), file(relset_1,k5_relset_1), []). fof(redefinition_k6_pencil_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v15_pencil_1(B) & m1_pboole(B,A) ) => k6_pencil_1(A,B) = k5_pencil_1(A,B) ) ), file(pencil_1,k6_pencil_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(redefinition_r1_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ( r1_subset_1(A,B) <=> r1_xboole_0(A,B) ) ) ), file(subset_1,r1_subset_1), []). fof(redefinition_r2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ( r2_subset_1(A,B) <=> r1_xboole_0(A,B) ) ) ), file(subset_1,r2_subset_1), []). fof(redefinition_r6_pboole,axiom,( ! [A,B,C] : ( ( m1_pboole(B,A) & m1_pboole(C,A) ) => ( r6_pboole(A,B,C) <=> B = C ) ) ), file(pboole,r6_pboole), []). fof(reflexivity_r1_tarski,axiom,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), []). fof(reflexivity_r1_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => r1_xreal_0(A,A) ) ), file(xreal_0,r1_xreal_0), []). fof(reflexivity_r2_pboole,axiom,( ! [A,B,C] : ( ( m1_pboole(B,A) & m1_pboole(C,A) ) => r2_pboole(A,B,B) ) ), file(pboole,r2_pboole), []). fof(reflexivity_r6_pboole,axiom,( ! [A,B,C] : ( ( m1_pboole(B,A) & m1_pboole(C,A) ) => r6_pboole(A,B,B) ) ), file(pboole,r6_pboole), []). fof(s1_nat_1__e4_25_1__pencil_3,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_finset_1(A) & v15_pencil_1(B) & m1_pboole(B,A) ) => ( ( ! [C] : ( ( ~ v13_pencil_1(C) & v16_pencil_1(C,A) & m4_pboole(C,A,k12_pralg_1(A,B)) ) => ! [D] : ( ( ~ v13_pencil_1(D) & v16_pencil_1(D,A) & m4_pboole(D,A,k12_pralg_1(A,B)) ) => ~ ( k3_pencil_1(A,C) = k3_pencil_1(A,D) & m1_pencil_2(k4_card_3(C),A,B) & m1_pencil_2(k4_card_3(D),A,B) & r1_subset_1(k4_card_3(C),k4_card_3(D)) & k1_pencil_3(A,C,D) = 0 & ! [E] : ( m2_finseq_1(E,k1_zfmisc_1(u1_struct_0(k6_pencil_1(A,B)))) => ~ ( k1_funct_1(E,1) = k4_card_3(C) & k1_funct_1(E,k3_finseq_1(E)) = k4_card_3(D) & ! [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) => ( r2_hidden(F,k4_finseq_1(E)) => m1_pencil_2(k1_funct_1(E,F),A,B) ) ) & ! [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) => ( r1_xreal_0(1,F) => ( r1_xreal_0(k3_finseq_1(E),F) | ! [G] : ( m1_pencil_2(G,A,B) => ! [H] : ( m1_pencil_2(H,A,B) => ( ( G = k1_funct_1(E,F) & H = k1_funct_1(E,k1_nat_1(F,1)) ) => ( r1_xboole_0(G,H) & r1_pencil_3(A,B,G,H) ) ) ) ) ) ) ) ) ) ) ) ) & ! [I] : ( m2_subset_1(I,k1_numbers,k5_numbers) => ( ! [J] : ( ( ~ v13_pencil_1(J) & v16_pencil_1(J,A) & m4_pboole(J,A,k12_pralg_1(A,B)) ) => ! [K] : ( ( ~ v13_pencil_1(K) & v16_pencil_1(K,A) & m4_pboole(K,A,k12_pralg_1(A,B)) ) => ~ ( k3_pencil_1(A,J) = k3_pencil_1(A,K) & m1_pencil_2(k4_card_3(J),A,B) & m1_pencil_2(k4_card_3(K),A,B) & r1_subset_1(k4_card_3(J),k4_card_3(K)) & k1_pencil_3(A,J,K) = I & ! [L] : ( m2_finseq_1(L,k1_zfmisc_1(u1_struct_0(k6_pencil_1(A,B)))) => ~ ( k1_funct_1(L,1) = k4_card_3(J) & k1_funct_1(L,k3_finseq_1(L)) = k4_card_3(K) & ! [M] : ( m2_subset_1(M,k1_numbers,k5_numbers) => ( r2_hidden(M,k4_finseq_1(L)) => m1_pencil_2(k1_funct_1(L,M),A,B) ) ) & ! [M] : ( m2_subset_1(M,k1_numbers,k5_numbers) => ( r1_xreal_0(1,M) => ( r1_xreal_0(k3_finseq_1(L),M) | ! [N] : ( m1_pencil_2(N,A,B) => ! [O] : ( m1_pencil_2(O,A,B) => ( ( N = k1_funct_1(L,M) & O = k1_funct_1(L,k1_nat_1(M,1)) ) => ( r1_xboole_0(N,O) & r1_pencil_3(A,B,N,O) ) ) ) ) ) ) ) ) ) ) ) ) => ! [P] : ( ( ~ v13_pencil_1(P) & v16_pencil_1(P,A) & m4_pboole(P,A,k12_pralg_1(A,B)) ) => ! [Q] : ( ( ~ v13_pencil_1(Q) & v16_pencil_1(Q,A) & m4_pboole(Q,A,k12_pralg_1(A,B)) ) => ~ ( k3_pencil_1(A,P) = k3_pencil_1(A,Q) & m1_pencil_2(k4_card_3(P),A,B) & m1_pencil_2(k4_card_3(Q),A,B) & r1_subset_1(k4_card_3(P),k4_card_3(Q)) & k1_pencil_3(A,P,Q) = k1_nat_1(I,1) & ! [R] : ( m2_finseq_1(R,k1_zfmisc_1(u1_struct_0(k6_pencil_1(A,B)))) => ~ ( k1_funct_1(R,1) = k4_card_3(P) & k1_funct_1(R,k3_finseq_1(R)) = k4_card_3(Q) & ! [S] : ( m2_subset_1(S,k1_numbers,k5_numbers) => ( r2_hidden(S,k4_finseq_1(R)) => m1_pencil_2(k1_funct_1(R,S),A,B) ) ) & ! [S] : ( m2_subset_1(S,k1_numbers,k5_numbers) => ( r1_xreal_0(1,S) => ( r1_xreal_0(k3_finseq_1(R),S) | ! [T] : ( m1_pencil_2(T,A,B) => ! [U] : ( m1_pencil_2(U,A,B) => ( ( T = k1_funct_1(R,S) & U = k1_funct_1(R,k1_nat_1(S,1)) ) => ( r1_xboole_0(T,U) & r1_pencil_3(A,B,T,U) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) => ! [I] : ( m2_subset_1(I,k1_numbers,k5_numbers) => ! [V] : ( ( ~ v13_pencil_1(V) & v16_pencil_1(V,A) & m4_pboole(V,A,k12_pralg_1(A,B)) ) => ! [W] : ( ( ~ v13_pencil_1(W) & v16_pencil_1(W,A) & m4_pboole(W,A,k12_pralg_1(A,B)) ) => ~ ( k3_pencil_1(A,V) = k3_pencil_1(A,W) & m1_pencil_2(k4_card_3(V),A,B) & m1_pencil_2(k4_card_3(W),A,B) & r1_subset_1(k4_card_3(V),k4_card_3(W)) & k1_pencil_3(A,V,W) = I & ! [X] : ( m2_finseq_1(X,k1_zfmisc_1(u1_struct_0(k6_pencil_1(A,B)))) => ~ ( k1_funct_1(X,1) = k4_card_3(V) & k1_funct_1(X,k3_finseq_1(X)) = k4_card_3(W) & ! [Y] : ( m2_subset_1(Y,k1_numbers,k5_numbers) => ( r2_hidden(Y,k4_finseq_1(X)) => m1_pencil_2(k1_funct_1(X,Y),A,B) ) ) & ! [Y] : ( m2_subset_1(Y,k1_numbers,k5_numbers) => ( r1_xreal_0(1,Y) => ( r1_xreal_0(k3_finseq_1(X),Y) | ! [Z] : ( m1_pencil_2(Z,A,B) => ! [A1] : ( m1_pencil_2(A1,A,B) => ( ( Z = k1_funct_1(X,Y) & A1 = k1_funct_1(X,k1_nat_1(Y,1)) ) => ( r1_xboole_0(Z,A1) & r1_pencil_3(A,B,Z,A1) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ), file(pencil_3,s1_nat_1__e4_25_1__pencil_3), []). fof(s1_nat_1__e9_25__pencil_3,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_finset_1(A) & v15_pencil_1(B) & m1_pboole(B,A) & ~ v13_pencil_1(C) & v16_pencil_1(C,A) & m4_pboole(C,A,k12_pralg_1(A,B)) & m2_finseq_1(D,k1_zfmisc_1(u1_struct_0(k6_pencil_1(A,B)))) ) => ( ( ! [E] : ( ( ~ v13_pencil_1(E) & v16_pencil_1(E,A) & m4_pboole(E,A,k12_pralg_1(A,B)) ) => ( k4_card_3(E) = k1_funct_1(D,0) => k3_pencil_1(A,C) = k3_pencil_1(A,E) ) ) & ! [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) => ( ! [G] : ( ( ~ v13_pencil_1(G) & v16_pencil_1(G,A) & m4_pboole(G,A,k12_pralg_1(A,B)) ) => ( k4_card_3(G) = k1_funct_1(D,F) => k3_pencil_1(A,C) = k3_pencil_1(A,G) ) ) => ! [H] : ( ( ~ v13_pencil_1(H) & v16_pencil_1(H,A) & m4_pboole(H,A,k12_pralg_1(A,B)) ) => ( k4_card_3(H) = k1_funct_1(D,k1_nat_1(F,1)) => k3_pencil_1(A,C) = k3_pencil_1(A,H) ) ) ) ) ) => ! [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) => ! [I] : ( ( ~ v13_pencil_1(I) & v16_pencil_1(I,A) & m4_pboole(I,A,k12_pralg_1(A,B)) ) => ( k4_card_3(I) = k1_funct_1(D,F) => k3_pencil_1(A,C) = k3_pencil_1(A,I) ) ) ) ) ) ), file(pencil_3,s1_nat_1__e9_25__pencil_3), []). fof(spc0_boole,axiom,( v1_xboole_0(0) ), file(boole,spc0_boole), []). fof(spc1_boole,axiom,( ~ v1_xboole_0(1) ), file(boole,spc1_boole), []). fof(spc1_numerals,axiom, ( v2_xreal_0(1) & m1_subset_1(1,k5_numbers) ), file(numerals,spc1_numerals), []). fof(spc2_boole,axiom,( ~ v1_xboole_0(2) ), file(boole,spc2_boole), []). fof(spc2_numerals,axiom, ( v2_xreal_0(2) & m1_subset_1(2,k5_numbers) ), file(numerals,spc2_numerals), []). fof(symmetry_r1_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ( r1_subset_1(A,B) => r1_subset_1(B,A) ) ) ), file(subset_1,r1_subset_1), []). fof(symmetry_r1_xboole_0,axiom,( ! [A,B] : ( r1_xboole_0(A,B) => r1_xboole_0(B,A) ) ), file(xboole_0,r1_xboole_0), []). fof(symmetry_r2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ( r2_subset_1(A,B) => r2_subset_1(B,A) ) ) ), file(subset_1,r2_subset_1), []). fof(symmetry_r6_pboole,axiom,( ! [A,B,C] : ( ( m1_pboole(B,A) & m1_pboole(C,A) ) => ( r6_pboole(A,B,C) => r6_pboole(A,C,B) ) ) ), file(pboole,r6_pboole), []). fof(t10_pencil_3,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( v11_pencil_1(B) & v14_pencil_1(B) & m1_pboole(B,A) ) => ! [C] : ( ( ~ v13_pencil_1(C) & v16_pencil_1(C,A) & m4_pboole(C,A,k12_pralg_1(A,B)) ) => ( m1_pencil_2(k4_card_3(C),A,B) => k1_funct_1(C,k3_pencil_1(A,C)) = k2_pre_topc(k1_pencil_1(A,B,k3_pencil_1(A,C))) ) ) ) ) ), file(pencil_3,t10_pencil_3), []). fof(t11_pencil_3,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( v11_pencil_1(B) & v14_pencil_1(B) & m1_pboole(B,A) ) => ! [C] : ( ( ~ v13_pencil_1(C) & v16_pencil_1(C,A) & m4_pboole(C,A,k12_pralg_1(A,B)) ) => ! [D] : ( ( ~ v13_pencil_1(D) & v16_pencil_1(D,A) & m4_pboole(D,A,k12_pralg_1(A,B)) ) => ( ( m1_pencil_2(k4_card_3(C),A,B) & m1_pencil_2(k4_card_3(D),A,B) & k3_pencil_1(A,C) = k3_pencil_1(A,D) ) => ( r2_subset_1(k4_card_3(C),k4_card_3(D)) | r6_pboole(A,C,D) ) ) ) ) ) ) ), file(pencil_3,t11_pencil_3), []). fof(t12_funct_1,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( r2_hidden(A,k1_relat_1(B)) => r2_hidden(k1_funct_1(B,A),k2_relat_1(B)) ) ) ), file(funct_1,t12_funct_1), []). fof(t12_pencil_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( ~ v13_pencil_1(B) & v16_pencil_1(B,A) & m1_pboole(B,A) ) => ! [C] : ( m1_subset_1(C,A) => ( C != k3_pencil_1(A,B) => ( ~ v1_xboole_0(k1_funct_1(B,C)) & v1_realset1(k1_funct_1(B,C)) ) ) ) ) ) ), file(pencil_1,t12_pencil_1), []). fof(t13_graph_2,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) ) => ( A != k1_xboole_0 => k1_nat_1(k3_finseq_1(k3_graph_2(B,A)),1) = k1_nat_1(k3_finseq_1(B),k3_finseq_1(A)) ) ) ) ), file(graph_2,t13_graph_2), []). fof(t13_pencil_3,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( v15_pencil_1(B) & m1_pboole(B,A) ) => ! [C] : ( m1_subset_1(C,A) => ! [D] : ( m1_subset_1(D,u1_struct_0(k2_pencil_1(A,B,C))) => ! [E] : ( ( ~ v13_pencil_1(E) & v16_pencil_1(E,A) & m4_pboole(E,A,k12_pralg_1(A,B)) ) => ( C != k3_pencil_1(A,E) => ( ~ v13_pencil_1(k2_polynom1(A,E,C,k1_struct_0(k2_pencil_1(A,B,C),D))) & v16_pencil_1(k2_polynom1(A,E,C,k1_struct_0(k2_pencil_1(A,B,C),D)),A) & m4_pboole(k2_polynom1(A,E,C,k1_struct_0(k2_pencil_1(A,B,C),D)),A,k12_pralg_1(A,B)) ) ) ) ) ) ) ) ), file(pencil_3,t13_pencil_3), []). fof(t14_graph_2,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(A)) ) => k1_funct_1(k3_graph_2(A,B),C) = k1_funct_1(A,C) ) ) ) ) ), file(graph_2,t14_graph_2), []). fof(t14_pencil_3,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( v15_pencil_1(B) & m1_pboole(B,A) ) => ! [C] : ( m1_subset_1(C,A) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k2_pencil_1(A,B,C)))) => ! [E] : ( ( ~ v13_pencil_1(E) & v16_pencil_1(E,A) & m4_pboole(E,A,k12_pralg_1(A,B)) ) => m1_subset_1(k4_card_3(k2_polynom1(A,E,C,D)),k1_zfmisc_1(u1_struct_0(k6_pencil_1(A,B)))) ) ) ) ) ) ), file(pencil_3,t14_pencil_3), []). fof(t15_graph_2,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(k3_finseq_1(A),C) | k1_funct_1(k3_graph_2(B,A),k1_nat_1(k3_finseq_1(B),C)) = k1_funct_1(A,k1_nat_1(C,1)) ) ) ) ) ) ), file(graph_2,t15_graph_2), []). fof(t16_graph_2,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) ) => ( ~ r1_xreal_0(k3_finseq_1(A),1) => k1_funct_1(k3_graph_2(B,A),k3_finseq_1(k3_graph_2(B,A))) = k1_funct_1(A,k3_finseq_1(A)) ) ) ) ), file(graph_2,t16_graph_2), []). fof(t17_graph_2,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) ) => r1_tarski(k2_relat_1(k3_graph_2(A,B)),k2_xboole_0(k2_relat_1(A),k2_relat_1(B))) ) ) ), file(graph_2,t17_graph_2), []). fof(t18_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => r1_xreal_0(0,A) ) ), file(nat_1,t18_nat_1), []). fof(t19_card_2,axiom, ( 0 = k1_card_1(k1_xboole_0) & 1 = k1_card_1(k4_ordinal2) & 2 = k1_card_1(k1_ordinal1(k4_ordinal2)) ), file(card_2,t19_card_2), []). fof(t19_pencil_3,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_finset_1(A) ) => ! [B] : ( m1_pboole(B,A) => ! [C] : ( m1_pboole(C,A) => ! [D] : ( m1_subset_1(D,A) => ( k1_funct_1(B,D) != k1_funct_1(C,D) => k1_pencil_3(A,B,C) = k1_nat_1(k1_pencil_3(A,B,k2_polynom1(A,C,D,k1_funct_1(B,D))),1) ) ) ) ) ) ), file(pencil_3,t19_pencil_3), []). 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(t1_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & r1_xreal_0(B,A) ) => A = B ) ) ) ), file(xreal_1,t1_xreal_1), []). fof(t21_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ~ r1_xreal_0(k2_xcmplx_0(A,1),0) ) ), file(nat_1,t21_nat_1), []). fof(t21_pencil_3,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( v15_pencil_1(B) & m1_pboole(B,A) ) => ! [C] : ( m1_pencil_2(C,A,B) => ! [D] : ( m1_pencil_2(D,A,B) => ( r1_xboole_0(C,D) => ( r1_pencil_3(A,B,C,D) <=> ! [E] : ( ( ~ v13_pencil_1(E) & v16_pencil_1(E,A) & m4_pboole(E,A,k12_pralg_1(A,B)) ) => ! [F] : ( ( ~ v13_pencil_1(F) & v16_pencil_1(F,A) & m4_pboole(F,A,k12_pralg_1(A,B)) ) => ( ( C = k4_card_3(E) & D = k4_card_3(F) ) => ( k3_pencil_1(A,E) = k3_pencil_1(A,F) & ? [G] : ( m1_subset_1(G,A) & G != k3_pencil_1(A,E) & ! [H] : ( m1_subset_1(H,A) => ( H != G => k1_funct_1(E,H) = k1_funct_1(F,H) ) ) & ! [H] : ( m1_subset_1(H,u1_struct_0(k2_pencil_1(A,B,G))) => ! [I] : ( m1_subset_1(I,u1_struct_0(k2_pencil_1(A,B,G))) => ( ( k1_funct_1(E,G) = k1_struct_0(k2_pencil_1(A,B,G),H) & k1_funct_1(F,G) = k1_struct_0(k2_pencil_1(A,B,G),I) ) => r1_pencil_1(k2_pencil_1(A,B,G),H,I) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ), file(pencil_3,t21_pencil_3), []). fof(t22_pencil_3,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_finset_1(A) ) => ! [B] : ( ( v15_pencil_1(B) & m1_pboole(B,A) ) => ( ! [C] : ( m1_subset_1(C,A) => v9_pencil_1(k2_pencil_1(A,B,C)) ) => ! [C] : ( m1_subset_1(C,A) => ! [D] : ( m1_subset_1(D,u1_struct_0(k2_pencil_1(A,B,C))) => ! [E] : ( ( ~ v13_pencil_1(E) & v16_pencil_1(E,A) & m4_pboole(E,A,k12_pralg_1(A,B)) ) => ! [F] : ( ( ~ v13_pencil_1(F) & v16_pencil_1(F,A) & m4_pboole(F,A,k12_pralg_1(A,B)) ) => ~ ( m1_pencil_2(k4_card_3(E),A,B) & m1_pencil_2(k4_card_3(F),A,B) & r6_pboole(A,E,k2_polynom1(A,F,C,k1_struct_0(k2_pencil_1(A,B,C),D))) & ~ r2_hidden(D,k1_funct_1(F,C)) & ! [G] : ( m2_finseq_1(G,k1_zfmisc_1(u1_struct_0(k6_pencil_1(A,B)))) => ~ ( k1_funct_1(G,1) = k4_card_3(E) & k1_funct_1(G,k3_finseq_1(G)) = k4_card_3(F) & ! [H] : ( m2_subset_1(H,k1_numbers,k5_numbers) => ( r2_hidden(H,k4_finseq_1(G)) => m1_pencil_2(k1_funct_1(G,H),A,B) ) ) & ! [H] : ( m2_subset_1(H,k1_numbers,k5_numbers) => ( r1_xreal_0(1,H) => ( r1_xreal_0(k3_finseq_1(G),H) | ! [I] : ( m1_pencil_2(I,A,B) => ! [J] : ( m1_pencil_2(J,A,B) => ( ( I = k1_funct_1(G,H) & J = k1_funct_1(G,k1_nat_1(H,1)) ) => ( r1_xboole_0(I,J) & r1_pencil_3(A,B,I,J) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ), file(pencil_3,t22_pencil_3), []). fof(t25_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( k3_finseq_1(A) = 0 <=> A = k1_xboole_0 ) ) ), file(finseq_1,t25_finseq_1), []). fof(t26_finseq_3,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ~ ( r2_hidden(B,k4_finseq_1(A)) & B = 0 ) ) ), file(finseq_3,t26_finseq_3), []). 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(t28_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ~ ( r1_xreal_0(A,B) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => B != k2_xcmplx_0(A,C) ) ) ) ) ), file(nat_1,t28_nat_1), []). fof(t29_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => r1_xreal_0(A,k2_xcmplx_0(A,B)) ) ) ), file(nat_1,t29_nat_1), []). fof(t2_pencil_3,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m1_subset_1(B,A) => ! [C] : ( ( v2_pralg_1(C) & v14_pencil_1(C) & m1_pboole(C,A) ) => ~ v3_realset2(k10_pralg_1(A,C,B)) ) ) ) ), file(pencil_3,t2_pencil_3), []). fof(t2_pua2mss1,axiom,( ! [A] : ( ( v1_relat_1(A) & v2_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v2_relat_1(B) & v1_funct_1(B) ) => ( k4_card_3(A) = k4_card_3(B) => A = B ) ) ) ), file(pua2mss1,t2_pua2mss1), []). fof(t2_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & v3_xreal_0(B) ) => v3_xreal_0(A) ) ) ) ), file(real,t2_real), []). fof(t2_subset,axiom,( ! [A,B] : ( m1_subset_1(A,B) => ( v1_xboole_0(B) | r2_hidden(A,B) ) ) ), file(subset,t2_subset), []). fof(t2_tarski,axiom,( ! [A,B] : ( ! [C] : ( r2_hidden(C,A) <=> r2_hidden(C,B) ) => A = B ) ), file(tarski,t2_tarski), []). fof(t2_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ( r1_xreal_0(A,B) & r1_xreal_0(B,C) ) => r1_xreal_0(A,C) ) ) ) ) ), file(xreal_1,t2_xreal_1), []). fof(t33_funct_7,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B,C] : ( r2_hidden(C,k1_relat_1(A)) => k1_funct_1(k2_funct_7(A,C,B),C) = B ) ) ), file(funct_7,t33_funct_7), []). fof(t34_funct_7,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B,C,D] : ( C != D => k1_funct_1(k2_funct_7(A,C,B),D) = k1_funct_1(A,D) ) ) ), file(funct_7,t34_funct_7), []). fof(t37_zfmisc_1,axiom,( ! [A,B] : ( r1_tarski(k1_tarski(A),B) <=> r2_hidden(A,B) ) ), file(zfmisc_1,t37_zfmisc_1), []). fof(t38_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( ~ r1_xreal_0(k2_xcmplx_0(B,1),A) <=> r1_xreal_0(A,B) ) ) ) ), file(nat_1,t38_nat_1), []). fof(t3_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( r1_xreal_0(A,B) & ~ v3_xreal_0(A) & v3_xreal_0(B) ) ) ) ), file(real,t3_real), []). fof(t3_subset,axiom,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), []). fof(t4_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k6_xcmplx_0(A,0) = A ) ), file(arithm,t4_arithm), []). fof(t4_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( r1_xreal_0(A,B) & ~ v2_xreal_0(B) & v2_xreal_0(A) ) ) ) ), file(real,t4_real), []). fof(t4_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(t59_card_2,axiom,( ! [A] : ( k1_card_1(A) = 0 => A = k1_xboole_0 ) ), file(card_2,t59_card_2), []). fof(t5_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(B) | v3_xreal_0(A) | v2_xreal_0(B) ) ) ) ) ), file(real,t5_real), []). fof(t5_subset,axiom,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), []). fof(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_pencil_3,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( v2_pralg_1(B) & m1_pboole(B,A) ) => ! [C] : ( m1_subset_1(C,A) => k1_funct_1(k12_pralg_1(A,B),C) = k2_pre_topc(k10_pralg_1(A,B,C)) ) ) ) ), file(pencil_3,t7_pencil_3), []). fof(t7_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,B) & ~ v2_xreal_0(A) & ~ v3_xreal_0(B) ) ) ) ), file(real,t7_real), []). fof(t8_boole,axiom,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), []). fof(t8_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,B) & ~ v3_xreal_0(B) & ~ v2_xreal_0(A) ) ) ) ), file(real,t8_real), []). fof(t8_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( r1_xreal_0(A,B) <=> r1_xreal_0(k2_xcmplx_0(A,C),k2_xcmplx_0(B,C)) ) ) ) ) ), file(xreal_1,t8_xreal_1), []). fof(t9_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( ( k1_relat_1(A) = k1_relat_1(B) & ! [C] : ( r2_hidden(C,k1_relat_1(A)) => k1_funct_1(A,C) = k1_funct_1(B,C) ) ) => A = B ) ) ) ), file(funct_1,t9_funct_1), []).