% Mizar problem: t31_necklace,necklace,1734,24 fof(t31_necklace,conjecture,( ! [A] : ( v4_ordinal2(A) => ~ ( r5_waybel_1(k4_necklace(A),k3_necklace(k4_necklace(A))) & A != 0 & A != 1 & A != 4 ) ) ), inference(mizar_bg_added,[status(thm)],[abstractness_v1_orders_2,antisymmetry_r2_hidden,cc1_afinsq_1,cc1_amistd_1,cc1_arytm_3,cc1_finset_1,cc1_funct_1,cc1_funct_2,cc1_necklace,cc1_orders_3,cc1_relset_1,cc1_tex_2,cc1_xreal_0,cc2_arytm_3,cc2_finset_1,cc2_funct_1,cc2_funct_2,cc2_necklace,cc2_tex_2,cc2_xreal_0,cc3_arytm_3,cc3_funct_2,cc3_tex_2,cc3_xreal_0,cc4_funct_2,cc4_tex_2,cc4_xreal_0,cc5_funct_2,cc5_tex_2,cc5_xreal_0,cc6_funct_2,cc6_tex_2,cc6_xreal_0,cc7_tex_2,cc7_xreal_0,cc8_xreal_0,commutativity_k1_nat_1,commutativity_k1_relset_1,commutativity_k2_nat_1,commutativity_k2_tarski,commutativity_k2_xboole_0,commutativity_k2_xcmplx_0,commutativity_k3_real_1,commutativity_k3_xcmplx_0,commutativity_k4_real_1,connectedness_r1_xreal_0,d10_necklace,d10_relat_1,d10_xboole_0,d1_bvfunc24,d1_enumset1,d1_quin_1,d1_relat_1,d1_xboole_0,d21_ordinal2,d2_enumset1,d2_tarski,d2_xboole_0,d3_enumset1,d3_square_1,d3_tarski,d4_xboole_0,d5_subset_1,d5_tarski,d7_necklace,d8_necklace,d9_necklace,dt_g1_orders_2,dt_k1_bvfunc24,dt_k1_card_1,dt_k1_enumset1,dt_k1_nat_1,dt_k1_necklace,dt_k1_numbers,dt_k1_quin_1,dt_k1_real_1,dt_k1_relset_1,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_enumset1,dt_k2_nat_1,dt_k2_necklace,dt_k2_quin_1,dt_k2_tarski,dt_k2_xboole_0,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k3_enumset1,dt_k3_necklace,dt_k3_real_1,dt_k3_subset_1,dt_k3_xcmplx_0,dt_k4_card_1,dt_k4_necklace,dt_k4_real_1,dt_k4_relat_1,dt_k4_tarski,dt_k4_xboole_0,dt_k4_xcmplx_0,dt_k5_numbers,dt_k5_ordinal2,dt_k5_real_1,dt_k5_square_1,dt_k6_partfun1,dt_k6_real_1,dt_k6_relat_1,dt_k6_relset_1,dt_k6_subset_1,dt_k6_xcmplx_0,dt_k7_square_1,dt_k7_xcmplx_0,dt_k8_square_1,dt_k9_square_1,dt_l1_orders_2,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_orders_2,dt_u1_struct_0,existence_l1_orders_2,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,fc10_xreal_0,fc11_xreal_0,fc12_finset_1,fc12_xreal_0,fc13_xreal_0,fc14_finset_1,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc1_finset_1,fc1_hilbert3,fc1_numbers,fc1_orders_2,fc1_orders_3,fc1_ordinal2,fc1_square_1,fc1_struct_0,fc1_subset_1,fc1_tex_2,fc1_xreal_0,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc23_xreal_0,fc24_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc2_afinsq_1,fc2_finset_1,fc2_funct_1,fc2_orders_2,fc2_orders_3,fc2_square_1,fc2_subset_1,fc30_xreal_0,fc3_finset_1,fc3_funct_1,fc3_necklace,fc3_orders_2,fc3_subset_1,fc3_xreal_0,fc4_finset_1,fc4_necklace,fc4_subset_1,fc4_xreal_0,fc5_finset_1,fc5_necklace,fc5_xreal_0,fc6_necklace,fc6_xreal_0,fc7_necklace,fc7_xreal_0,fc8_necklace,fc8_xreal_0,fc9_finset_1,fc9_necklace,fc9_xreal_0,fraenkel_a_0_5_necklace,fraenkel_a_0_6_necklace,fraenkel_a_0_7_necklace,fraenkel_a_0_8_necklace,fraenkel_a_1_1_necklace,fraenkel_a_1_3_necklace,free_g1_orders_2,idempotence_k1_relset_1,idempotence_k2_xboole_0,involutiveness_k1_real_1,involutiveness_k3_subset_1,involutiveness_k4_relat_1,involutiveness_k4_xcmplx_0,involutiveness_k6_relset_1,rc1_afinsq_1,rc1_amistd_1,rc1_arytm_3,rc1_finset_1,rc1_funcop_1,rc1_funct_1,rc1_funct_2,rc1_necklace,rc1_orders_2,rc1_orders_3,rc1_subset_1,rc1_tex_2,rc1_xreal_0,rc2_finset_1,rc2_funct_1,rc2_funct_2,rc2_orders_2,rc2_orders_3,rc2_subset_1,rc2_tex_2,rc2_xreal_0,rc3_afinsq_1,rc3_finset_1,rc3_funct_1,rc3_struct_0,rc3_tex_2,rc3_xreal_0,rc4_afinsq_1,rc4_finset_1,rc4_funct_1,rc4_tex_2,rc4_xreal_0,rc5_struct_0,rc5_tex_2,rc6_tex_2,rc7_tex_2,redefinition_k1_nat_1,redefinition_k1_real_1,redefinition_k1_relset_1,redefinition_k2_nat_1,redefinition_k2_quin_1,redefinition_k3_real_1,redefinition_k4_card_1,redefinition_k4_real_1,redefinition_k5_numbers,redefinition_k5_real_1,redefinition_k6_partfun1,redefinition_k6_real_1,redefinition_k6_relset_1,redefinition_k6_subset_1,redefinition_k7_square_1,redefinition_k9_square_1,redefinition_m2_relset_1,redefinition_m2_subset_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,reflexivity_r5_waybel_1,spc0_boole,spc1_boole,spc1_numerals,spc2_boole,spc2_numerals,spc3_boole,spc3_numerals,spc4_boole,spc4_numerals,spc5_boole,spc5_numerals,spc9_boole,spc9_numerals,symmetry_r1_xboole_0,t15_quin_1,t18_necklace,t19_nat_1,t19_necklace,t1_arithm,t1_boole,t1_card_5,t1_numerals,t1_real,t1_subset,t1_yellow11,t20_necklace,t21_necklace,t25_mcart_1,t28_necklace,t2_arithm,t2_cqc_the1,t2_real,t2_subset,t2_tarski,t33_zfmisc_1,t38_nat_1,t3_arithm,t3_boole,t3_necklace,t3_real,t3_subset,t3_xboole_0,t47_card_1,t4_arithm,t4_boole,t4_real,t4_subset,t5_arithm,t5_cqc_the1,t5_real,t5_subset,t63_card_2,t65_card_2,t66_card_1,t6_arithm,t6_boole,t6_necklace,t6_real,t76_card_2,t7_boole,t7_real,t86_xboole_1,t89_square_1,t8_boole,t8_real,t9_xreal_1]), [file(necklace,t31_necklace)]). fof(abstractness_v1_orders_2,axiom,( ! [A] : ( l1_orders_2(A) => ( v1_orders_2(A) => A = g1_orders_2(u1_struct_0(A),u1_orders_2(A)) ) ) ), file(orders_2,v1_orders_2), []). fof(antisymmetry_r2_hidden,axiom,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), []). fof(cc1_afinsq_1,axiom,( ! [A] : ( v4_ordinal2(A) => v1_finset_1(A) ) ), file(afinsq_1,cc1_afinsq_1), []). fof(cc1_amistd_1,axiom,( ! [A] : ( v1_relat_1(A) => ( v1_relat_1(A) & v1_setfam_1(A) ) ) ), file(amistd_1,cc1_amistd_1), []). 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_funct_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_funct_1(A) ) ), file(funct_1,cc1_funct_1), []). fof(cc1_funct_2,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => ( ( v1_funct_1(C) & v1_partfun1(C,A,B) ) => ( v1_funct_1(C) & v1_funct_2(C,A,B) ) ) ) ), file(funct_2,cc1_funct_2), []). fof(cc1_necklace,axiom,( ! [A] : ( v4_ordinal2(A) => v1_card_1(A) ) ), file(necklace,cc1_necklace), []). fof(cc1_orders_3,axiom,( ! [A] : ( l1_orders_2(A) => ( v3_struct_0(A) => v1_orders_3(A) ) ) ), file(orders_3,cc1_orders_3), []). 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_tex_2,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( ~ v1_tex_2(B,k1_zfmisc_1(A)) => ~ v1_xboole_0(B) ) ) ) ), file(tex_2,cc1_tex_2), []). 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_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ) ), file(funct_1,cc2_funct_1), []). fof(cc2_funct_2,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => ( ( v1_funct_1(C) & v1_funct_2(C,A,B) & v3_funct_2(C,A,B) ) => ( v1_funct_1(C) & v2_funct_1(C) & v1_funct_2(C,A,B) & v2_funct_2(C,A,B) ) ) ) ), file(funct_2,cc2_funct_2), []). fof(cc2_necklace,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_realset1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_setfam_1(A) ) ) ), file(necklace,cc2_necklace), []). fof(cc2_tex_2,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_xboole_0(B) => v1_tex_2(B,k1_zfmisc_1(A)) ) ) ) ), file(tex_2,cc2_tex_2), []). 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_funct_2,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => ( ( v1_funct_1(C) & v2_funct_1(C) & v1_funct_2(C,A,B) & v2_funct_2(C,A,B) ) => ( v1_funct_1(C) & v1_funct_2(C,A,B) & v3_funct_2(C,A,B) ) ) ) ), file(funct_2,cc3_funct_2), []). fof(cc3_tex_2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_realset1(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_tex_2(B,k1_zfmisc_1(A)) => ( v1_xboole_0(B) & v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) & v5_membered(B) & v1_tex_2(B,k1_zfmisc_1(A)) ) ) ) ) ), file(tex_2,cc3_tex_2), []). 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_funct_2,axiom,( ! [A,B] : ( m1_relset_1(B,A,A) => ( ( v1_funct_1(B) & v1_partfun1(B,A,A) & v1_relat_2(B) & v1_funct_2(B,A,A) ) => ( v1_funct_1(B) & v2_funct_1(B) & v1_funct_2(B,A,A) & v2_funct_2(B,A,A) & v3_funct_2(B,A,A) ) ) ) ), file(funct_2,cc4_funct_2), []). fof(cc4_tex_2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_realset1(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( ~ v1_xboole_0(B) => ( ~ v1_xboole_0(B) & ~ v1_tex_2(B,k1_zfmisc_1(A)) ) ) ) ) ), file(tex_2,cc4_tex_2), []). 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_funct_2,axiom,( ! [A,B] : ( ~ v1_xboole_0(B) => ! [C] : ( m1_relset_1(C,A,B) => ( ( v1_funct_1(C) & v1_funct_2(C,A,B) ) => ( v1_funct_1(C) & v1_partfun1(C,A,B) & v1_funct_2(C,A,B) ) ) ) ) ), file(funct_2,cc5_funct_2), []). fof(cc5_tex_2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_realset1(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( ( ~ v1_xboole_0(B) & ~ v1_tex_2(B,k1_zfmisc_1(A)) ) => ( ~ v1_xboole_0(B) & v1_realset1(B) & ~ v1_tex_2(B,k1_zfmisc_1(A)) ) ) ) ) ), file(tex_2,cc5_tex_2), []). 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_funct_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ! [C] : ( m1_relset_1(C,A,B) => ( ( v1_funct_1(C) & v1_funct_2(C,A,B) ) => ( v1_funct_1(C) & ~ v1_xboole_0(C) & v1_partfun1(C,A,B) & v1_funct_2(C,A,B) ) ) ) ) ), file(funct_2,cc6_funct_2), []). fof(cc6_tex_2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & ~ v1_realset1(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( ( ~ v1_xboole_0(B) & v1_realset1(B) ) => ( ~ v1_xboole_0(B) & v1_tex_2(B,k1_zfmisc_1(A)) ) ) ) ) ), file(tex_2,cc6_tex_2), []). 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_tex_2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & ~ v1_realset1(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( ( ~ v1_xboole_0(B) & ~ v1_tex_2(B,k1_zfmisc_1(A)) ) => ( ~ v1_xboole_0(B) & ~ v1_realset1(B) ) ) ) ) ), file(tex_2,cc7_tex_2), []). 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_k1_relset_1,axiom,( ! [A,B,C,D] : ( ( m1_relset_1(C,A,B) & m1_relset_1(D,A,B) ) => k1_relset_1(A,B,C,D) = k1_relset_1(A,B,D,C) ) ), file(relset_1,k1_relset_1), []). fof(commutativity_k2_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k2_nat_1(A,B) = k2_nat_1(B,A) ) ), file(nat_1,k2_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_k3_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k3_real_1(A,B) = k3_real_1(B,A) ) ), file(real_1,k3_real_1), []). fof(commutativity_k3_xcmplx_0,axiom,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k3_xcmplx_0(A,B) = k3_xcmplx_0(B,A) ) ), file(xcmplx_0,k3_xcmplx_0), []). fof(commutativity_k4_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k4_real_1(A,B) = k4_real_1(B,A) ) ), file(real_1,k4_real_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(d10_necklace,axiom,( ! [A] : ( v4_ordinal2(A) => k4_necklace(A) = k2_necklace(k1_necklace(A)) ) ), file(necklace,d10_necklace), []). fof(d10_relat_1,axiom,( ! [A,B] : ( v1_relat_1(B) => ( B = k6_relat_1(A) <=> ! [C,D] : ( r2_hidden(k4_tarski(C,D),B) <=> ( r2_hidden(C,A) & C = D ) ) ) ) ), file(relat_1,d10_relat_1), []). fof(d10_xboole_0,axiom,( ! [A,B] : ( A = B <=> ( r1_tarski(A,B) & r1_tarski(B,A) ) ) ), file(xboole_0,d10_xboole_0), []). fof(d1_bvfunc24,axiom,( ! [A,B,C,D,E,F,G,H,I,J] : ( J = k1_bvfunc24(A,B,C,D,E,F,G,H,I) <=> ! [K] : ( r2_hidden(K,J) <=> ~ ( K != A & K != B & K != C & K != D & K != E & K != F & K != G & K != H & K != I ) ) ) ), file(bvfunc24,d1_bvfunc24), []). 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_quin_1,axiom,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => ! [C] : ( v1_xcmplx_0(C) => k1_quin_1(A,B,C) = k6_xcmplx_0(k5_square_1(B),k3_xcmplx_0(k3_xcmplx_0(4,A),C)) ) ) ) ), file(quin_1,d1_quin_1), []). fof(d1_relat_1,axiom,( ! [A] : ( v1_relat_1(A) <=> ! [B] : ~ ( r2_hidden(B,A) & ! [C,D] : B != k4_tarski(C,D) ) ) ), file(relat_1,d1_relat_1), []). fof(d1_xboole_0,axiom,( ! [A] : ( A = k1_xboole_0 <=> ! [B] : ~ r2_hidden(B,A) ) ), file(xboole_0,d1_xboole_0), []). fof(d21_ordinal2,axiom,( ! [A] : ( v4_ordinal2(A) <=> r2_hidden(A,k5_ordinal2) ) ), file(ordinal2,d21_ordinal2), []). 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_tarski,axiom,( ! [A,B,C] : ( C = k2_tarski(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( D = A | D = B ) ) ) ), file(tarski,d2_tarski), []). 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_enumset1,axiom,( ! [A,B,C,D,E,F] : ( F = k3_enumset1(A,B,C,D,E) <=> ! [G] : ( r2_hidden(G,F) <=> ~ ( G != A & G != B & G != C & G != D & G != E ) ) ) ), file(enumset1,d3_enumset1), []). fof(d3_square_1,axiom,( ! [A] : ( v1_xcmplx_0(A) => k5_square_1(A) = k3_xcmplx_0(A,A) ) ), file(square_1,d3_square_1), []). 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_xboole_0,axiom,( ! [A,B,C] : ( C = k4_xboole_0(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( r2_hidden(D,A) & ~ r2_hidden(D,B) ) ) ) ), file(xboole_0,d4_xboole_0), []). fof(d5_subset_1,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => k3_subset_1(A,B) = k4_xboole_0(A,B) ) ), file(subset_1,d5_subset_1), []). fof(d5_tarski,axiom,( ! [A,B] : k4_tarski(A,B) = k2_tarski(k2_tarski(A,B),k1_tarski(A)) ), file(tarski,d5_tarski), []). fof(d7_necklace,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( ( v1_orders_2(B) & l1_orders_2(B) ) => ( B = k1_necklace(A) <=> ( u1_struct_0(B) = A & u1_orders_2(B) = a_1_1_necklace(A) ) ) ) ) ), file(necklace,d7_necklace), []). fof(d8_necklace,axiom,( ! [A] : ( l1_orders_2(A) => ! [B] : ( ( v1_orders_2(B) & l1_orders_2(B) ) => ( B = k2_necklace(A) <=> ( u1_struct_0(B) = u1_struct_0(A) & u1_orders_2(B) = k1_relset_1(u1_struct_0(A),u1_struct_0(A),u1_orders_2(A),k6_relset_1(u1_struct_0(A),u1_struct_0(A),u1_orders_2(A))) ) ) ) ) ), file(necklace,d8_necklace), []). fof(d9_necklace,axiom,( ! [A] : ( l1_orders_2(A) => ! [B] : ( ( v1_orders_2(B) & l1_orders_2(B) ) => ( B = k3_necklace(A) <=> ( u1_struct_0(B) = u1_struct_0(A) & u1_orders_2(B) = k6_subset_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k3_subset_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_orders_2(A)),k6_partfun1(u1_struct_0(A))) ) ) ) ) ), file(necklace,d9_necklace), []). fof(dt_g1_orders_2,axiom,( ! [A,B] : ( m1_relset_1(B,A,A) => ( v1_orders_2(g1_orders_2(A,B)) & l1_orders_2(g1_orders_2(A,B)) ) ) ), file(orders_2,g1_orders_2), []). fof(dt_k1_bvfunc24,axiom,( $true ), file(bvfunc24,k1_bvfunc24), []). 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_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_necklace,axiom,( ! [A] : ( v4_ordinal2(A) => ( v1_orders_2(k1_necklace(A)) & l1_orders_2(k1_necklace(A)) ) ) ), file(necklace,k1_necklace), []). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), []). fof(dt_k1_quin_1,axiom,( $true ), file(quin_1,k1_quin_1), []). fof(dt_k1_real_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => m1_subset_1(k1_real_1(A),k1_numbers) ) ), file(real_1,k1_real_1), []). fof(dt_k1_relset_1,axiom,( ! [A,B,C,D] : ( ( m1_relset_1(C,A,B) & m1_relset_1(D,A,B) ) => m2_relset_1(k1_relset_1(A,B,C,D),A,B) ) ), file(relset_1,k1_relset_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_k2_enumset1,axiom,( $true ), file(enumset1,k2_enumset1), []). fof(dt_k2_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k2_nat_1(A,B),k1_numbers,k5_numbers) ) ), file(nat_1,k2_nat_1), []). fof(dt_k2_necklace,axiom,( ! [A] : ( l1_orders_2(A) => ( v1_orders_2(k2_necklace(A)) & l1_orders_2(k2_necklace(A)) ) ) ), file(necklace,k2_necklace), []). fof(dt_k2_quin_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) & m1_subset_1(C,k1_numbers) ) => m1_subset_1(k2_quin_1(A,B,C),k1_numbers) ) ), file(quin_1,k2_quin_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_enumset1,axiom,( $true ), file(enumset1,k3_enumset1), []). fof(dt_k3_necklace,axiom,( ! [A] : ( l1_orders_2(A) => ( v1_orders_2(k3_necklace(A)) & l1_orders_2(k3_necklace(A)) ) ) ), file(necklace,k3_necklace), []). fof(dt_k3_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k3_real_1(A,B),k1_numbers) ) ), file(real_1,k3_real_1), []). fof(dt_k3_subset_1,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => m1_subset_1(k3_subset_1(A,B),k1_zfmisc_1(A)) ) ), file(subset_1,k3_subset_1), []). fof(dt_k3_xcmplx_0,axiom,( $true ), file(xcmplx_0,k3_xcmplx_0), []). 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_necklace,axiom,( ! [A] : ( v4_ordinal2(A) => ( v1_orders_2(k4_necklace(A)) & l1_orders_2(k4_necklace(A)) ) ) ), file(necklace,k4_necklace), []). fof(dt_k4_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k4_real_1(A,B),k1_numbers) ) ), file(real_1,k4_real_1), []). fof(dt_k4_relat_1,axiom,( ! [A] : ( v1_relat_1(A) => v1_relat_1(k4_relat_1(A)) ) ), file(relat_1,k4_relat_1), []). fof(dt_k4_tarski,axiom,( $true ), file(tarski,k4_tarski), []). fof(dt_k4_xboole_0,axiom,( $true ), file(xboole_0,k4_xboole_0), []). fof(dt_k4_xcmplx_0,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_xcmplx_0(k4_xcmplx_0(A)) ) ), file(xcmplx_0,k4_xcmplx_0), []). fof(dt_k5_numbers,axiom,( m1_subset_1(k5_numbers,k1_zfmisc_1(k1_numbers)) ), file(numbers,k5_numbers), []). fof(dt_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), []). fof(dt_k5_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k5_real_1(A,B),k1_numbers) ) ), file(real_1,k5_real_1), []). fof(dt_k5_square_1,axiom,( $true ), file(square_1,k5_square_1), []). fof(dt_k6_partfun1,axiom,( ! [A] : ( v1_partfun1(k6_partfun1(A),A,A) & m2_relset_1(k6_partfun1(A),A,A) ) ), file(partfun1,k6_partfun1), []). fof(dt_k6_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k6_real_1(A,B),k1_numbers) ) ), file(real_1,k6_real_1), []). fof(dt_k6_relat_1,axiom,( ! [A] : v1_relat_1(k6_relat_1(A)) ), file(relat_1,k6_relat_1), []). fof(dt_k6_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => m2_relset_1(k6_relset_1(A,B,C),B,A) ) ), file(relset_1,k6_relset_1), []). fof(dt_k6_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(k6_subset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(subset_1,k6_subset_1), []). fof(dt_k6_xcmplx_0,axiom,( $true ), file(xcmplx_0,k6_xcmplx_0), []). fof(dt_k7_square_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => m1_subset_1(k7_square_1(A),k1_numbers) ) ), file(square_1,k7_square_1), []). fof(dt_k7_xcmplx_0,axiom,( $true ), file(xcmplx_0,k7_xcmplx_0), []). fof(dt_k8_square_1,axiom,( ! [A] : ( v1_xreal_0(A) => v1_xreal_0(k8_square_1(A)) ) ), file(square_1,k8_square_1), []). fof(dt_k9_square_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => m1_subset_1(k9_square_1(A),k1_numbers) ) ), file(square_1,k9_square_1), []). fof(dt_l1_orders_2,axiom,( ! [A] : ( l1_orders_2(A) => l1_struct_0(A) ) ), file(orders_2,l1_orders_2), []). fof(dt_l1_struct_0,axiom,( $true ), file(struct_0,l1_struct_0), []). 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_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_orders_2,axiom,( ! [A] : ( l1_orders_2(A) => m2_relset_1(u1_orders_2(A),u1_struct_0(A),u1_struct_0(A)) ) ), file(orders_2,u1_orders_2), []). fof(dt_u1_struct_0,axiom,( $true ), file(struct_0,u1_struct_0), []). fof(existence_l1_orders_2,axiom,( ? [A] : l1_orders_2(A) ), file(orders_2,l1_orders_2), []). fof(existence_l1_struct_0,axiom,( ? [A] : l1_struct_0(A) ), file(struct_0,l1_struct_0), []). 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_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_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_finset_1,axiom,( ! [A,B] : ( v1_finset_1(A) => v1_finset_1(k4_xboole_0(A,B)) ) ), file(finset_1,fc12_finset_1), []). 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_xreal_0,axiom,( ! [A] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) ) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & ~ v3_xreal_0(k4_xcmplx_0(A)) ) ) ), file(xreal_0,fc13_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(fc14_xreal_0,axiom,( ! [A] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & ~ v2_xreal_0(k4_xcmplx_0(A)) ) ) ), file(xreal_0,fc14_xreal_0), []). 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_finset_1,axiom,( ! [A] : ( ~ v1_xboole_0(k1_tarski(A)) & v1_finset_1(k1_tarski(A)) ) ), file(finset_1,fc1_finset_1), []). fof(fc1_hilbert3,axiom, ( v1_xboole_0(k1_xboole_0) & v1_relat_1(k1_xboole_0) & v3_relat_1(k1_xboole_0) & v1_funct_1(k1_xboole_0) & v2_funct_1(k1_xboole_0) & v1_funcop_1(k1_xboole_0) ), file(hilbert3,fc1_hilbert3), []). fof(fc1_numbers,axiom,( ~ v1_xboole_0(k1_numbers) ), file(numbers,fc1_numbers), []). fof(fc1_orders_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_relset_1(B,A,A) ) => ( ~ v3_struct_0(g1_orders_2(A,B)) & v1_orders_2(g1_orders_2(A,B)) ) ) ), file(orders_2,fc1_orders_2), []). fof(fc1_orders_3,axiom, ( v3_struct_0(g1_orders_2(k1_xboole_0,k6_partfun1(k1_xboole_0))) & v1_orders_2(g1_orders_2(k1_xboole_0,k6_partfun1(k1_xboole_0))) & v2_orders_2(g1_orders_2(k1_xboole_0,k6_partfun1(k1_xboole_0))) & v3_orders_2(g1_orders_2(k1_xboole_0,k6_partfun1(k1_xboole_0))) & v4_orders_2(g1_orders_2(k1_xboole_0,k6_partfun1(k1_xboole_0))) ), file(orders_3,fc1_orders_3), []). fof(fc1_ordinal2,axiom, ( v1_ordinal1(k5_ordinal2) & v2_ordinal1(k5_ordinal2) & v3_ordinal1(k5_ordinal2) & ~ v1_xboole_0(k5_ordinal2) ), file(ordinal2,fc1_ordinal2), []). fof(fc1_square_1,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_xcmplx_0(k5_square_1(A)) ) ), file(square_1,fc1_square_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_tex_2,axiom,( ! [A] : ( ~ v1_xboole_0(k1_tarski(A)) & v1_realset1(k1_tarski(A)) & v1_finset_1(k1_tarski(A)) ) ), file(tex_2,fc1_tex_2), []). fof(fc1_xreal_0,axiom,( ! [A] : ( v1_xreal_0(A) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) ) ) ), file(xreal_0,fc1_xreal_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(fc21_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & ~ v2_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc21_xreal_0), []). fof(fc22_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(B,A)) & v1_xreal_0(k3_xcmplx_0(B,A)) & ~ v2_xreal_0(k3_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc22_xreal_0), []). fof(fc23_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & ~ v3_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc23_xreal_0), []). fof(fc24_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & ~ v3_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc24_xreal_0), []). fof(fc27_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(A,B)) & v1_xreal_0(k7_xcmplx_0(A,B)) & ~ v2_xreal_0(k7_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc27_xreal_0), []). fof(fc28_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(B,A)) & v1_xreal_0(k7_xcmplx_0(B,A)) & ~ v2_xreal_0(k7_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc28_xreal_0), []). fof(fc29_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(A,B)) & v1_xreal_0(k7_xcmplx_0(A,B)) & ~ v3_xreal_0(k7_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc29_xreal_0), []). fof(fc2_afinsq_1,axiom, ( v1_xboole_0(k1_xboole_0) & v1_relat_1(k1_xboole_0) & v3_relat_1(k1_xboole_0) & v1_funct_1(k1_xboole_0) & v2_funct_1(k1_xboole_0) & v1_ordinal1(k1_xboole_0) & v2_ordinal1(k1_xboole_0) & v3_ordinal1(k1_xboole_0) & v5_ordinal1(k1_xboole_0) & v4_ordinal2(k1_xboole_0) & v1_xcmplx_0(k1_xboole_0) & v1_finset_1(k1_xboole_0) & v1_finseq_1(k1_xboole_0) & v1_funcop_1(k1_xboole_0) & v1_funct_7(k1_xboole_0) & v1_xreal_0(k1_xboole_0) & ~ v2_xreal_0(k1_xboole_0) & ~ v3_xreal_0(k1_xboole_0) ), file(afinsq_1,fc2_afinsq_1), []). 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_funct_1,axiom,( ! [A] : ( v1_relat_1(k6_relat_1(A)) & v1_funct_1(k6_relat_1(A)) ) ), file(funct_1,fc2_funct_1), []). fof(fc2_orders_2,axiom,( ! [A] : ( ( v2_orders_2(A) & v3_orders_2(A) & v4_orders_2(A) & l1_orders_2(A) ) => ( v1_relat_1(u1_orders_2(A)) & v1_relat_2(u1_orders_2(A)) & v4_relat_2(u1_orders_2(A)) & v8_relat_2(u1_orders_2(A)) & v1_partfun1(u1_orders_2(A),u1_struct_0(A),u1_struct_0(A)) ) ) ), file(orders_2,fc2_orders_2), []). fof(fc2_orders_3,axiom,( ! [A] : ( ( v3_struct_0(A) & l1_orders_2(A) ) => ( v1_relat_1(u1_orders_2(A)) & v1_funct_1(u1_orders_2(A)) & v2_funct_1(u1_orders_2(A)) & v1_xboole_0(u1_orders_2(A)) ) ) ), file(orders_3,fc2_orders_3), []). fof(fc2_square_1,axiom,( ! [A] : ( v1_xreal_0(A) => ( v1_xcmplx_0(k5_square_1(A)) & v1_xreal_0(k5_square_1(A)) ) ) ), file(square_1,fc2_square_1), []). fof(fc2_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_tarski(A)) ), file(subset_1,fc2_subset_1), []). fof(fc30_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(A,B)) & v1_xreal_0(k7_xcmplx_0(A,B)) & ~ v3_xreal_0(k7_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc30_xreal_0), []). fof(fc3_finset_1,axiom,( ! [A,B,C] : v1_finset_1(k1_enumset1(A,B,C)) ), file(finset_1,fc3_finset_1), []). fof(fc3_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) => ( v1_relat_1(k4_relat_1(A)) & v1_funct_1(k4_relat_1(A)) ) ) ), file(funct_1,fc3_funct_1), []). fof(fc3_necklace,axiom,( ! [A] : ( l1_orders_2(A) => ( v1_orders_2(k2_necklace(A)) & v1_necklace(k2_necklace(A)) ) ) ), file(necklace,fc3_necklace), []). fof(fc3_orders_2,axiom,( ! [A,B] : ( ( v1_relat_2(B) & v4_relat_2(B) & v8_relat_2(B) & v1_partfun1(B,A,A) & m1_relset_1(B,A,A) ) => ( v1_orders_2(g1_orders_2(A,B)) & v2_orders_2(g1_orders_2(A,B)) & v3_orders_2(g1_orders_2(A,B)) & v4_orders_2(g1_orders_2(A,B)) ) ) ), file(orders_2,fc3_orders_2), []). fof(fc3_subset_1,axiom,( ! [A,B] : ~ v1_xboole_0(k2_tarski(A,B)) ), file(subset_1,fc3_subset_1), []). fof(fc3_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc3_xreal_0), []). fof(fc4_finset_1,axiom,( ! [A,B,C,D] : v1_finset_1(k2_enumset1(A,B,C,D)) ), file(finset_1,fc4_finset_1), []). fof(fc4_necklace,axiom,( ! [A] : ( ( v1_necklace(A) & l1_orders_2(A) ) => ( v1_relat_1(u1_orders_2(A)) & v3_relat_2(u1_orders_2(A)) & v1_setfam_1(u1_orders_2(A)) ) ) ), file(necklace,fc4_necklace), []). 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(fc4_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc4_xreal_0), []). fof(fc5_finset_1,axiom,( ! [A,B,C,D,E] : v1_finset_1(k3_enumset1(A,B,C,D,E)) ), file(finset_1,fc5_finset_1), []). fof(fc5_necklace,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_orders_2(A) ) => ( ~ v3_struct_0(k3_necklace(A)) & v1_orders_2(k3_necklace(A)) ) ) ), file(necklace,fc5_necklace), []). 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(fc6_necklace,axiom,( ! [A] : ( v4_ordinal2(A) => ( v1_orders_2(k4_necklace(A)) & v1_necklace(k4_necklace(A)) ) ) ), file(necklace,fc6_necklace), []). fof(fc6_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(A,B)) & v1_xreal_0(k7_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc6_xreal_0), []). fof(fc7_necklace,axiom,( ! [A] : ( v4_ordinal2(A) => ( v1_orders_2(k4_necklace(A)) & v1_necklace(k4_necklace(A)) & v3_necklace(k4_necklace(A)) ) ) ), file(necklace,fc7_necklace), []). 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_necklace,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v4_ordinal2(A) ) => ( ~ v3_struct_0(k4_necklace(A)) & v1_orders_2(k4_necklace(A)) & v1_necklace(k4_necklace(A)) & v3_necklace(k4_necklace(A)) ) ) ), file(necklace,fc8_necklace), []). 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_necklace,axiom,( ! [A] : ( v4_ordinal2(A) => v1_finset_1(u1_struct_0(k4_necklace(A))) ) ), file(necklace,fc9_necklace), []). 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_0_5_necklace,axiom,( ! [A] : ( r2_hidden(A,a_0_5_necklace) <=> ? [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) & A = k4_tarski(B,k1_nat_1(B,1)) & ~ r1_xreal_0(2,k1_nat_1(B,1)) ) ) ), file(necklace,a_0_5_necklace), []). fof(fraenkel_a_0_6_necklace,axiom,( ! [A] : ( r2_hidden(A,a_0_6_necklace) <=> ? [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) & A = k4_tarski(k1_nat_1(B,1),B) & ~ r1_xreal_0(2,k1_nat_1(B,1)) ) ) ), file(necklace,a_0_6_necklace), []). fof(fraenkel_a_0_7_necklace,axiom,( ! [A] : ( r2_hidden(A,a_0_7_necklace) <=> ? [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) & A = k4_tarski(B,k1_nat_1(B,1)) & ~ r1_xreal_0(3,k1_nat_1(B,1)) ) ) ), file(necklace,a_0_7_necklace), []). fof(fraenkel_a_0_8_necklace,axiom,( ! [A] : ( r2_hidden(A,a_0_8_necklace) <=> ? [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) & A = k4_tarski(k1_nat_1(B,1),B) & ~ r1_xreal_0(3,k1_nat_1(B,1)) ) ) ), file(necklace,a_0_8_necklace), []). fof(fraenkel_a_1_1_necklace,axiom,( ! [A,B] : ( v4_ordinal2(B) => ( r2_hidden(A,a_1_1_necklace(B)) <=> ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & A = k4_tarski(C,k1_nat_1(C,1)) & ~ r1_xreal_0(B,k1_nat_1(C,1)) ) ) ) ), file(necklace,a_1_1_necklace), []). fof(fraenkel_a_1_3_necklace,axiom,( ! [A,B] : ( v4_ordinal2(B) => ( r2_hidden(A,a_1_3_necklace(B)) <=> ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & A = k4_tarski(k1_nat_1(C,1),C) & ~ r1_xreal_0(B,k1_nat_1(C,1)) ) ) ) ), file(necklace,a_1_3_necklace), []). fof(free_g1_orders_2,axiom,( ! [A,B] : ( m1_relset_1(B,A,A) => ! [C,D] : ( g1_orders_2(A,B) = g1_orders_2(C,D) => ( A = C & B = D ) ) ) ), file(orders_2,g1_orders_2), []). fof(idempotence_k1_relset_1,axiom,( ! [A,B,C,D] : ( ( m1_relset_1(C,A,B) & m1_relset_1(D,A,B) ) => k1_relset_1(A,B,C,C) = C ) ), file(relset_1,k1_relset_1), []). fof(idempotence_k2_xboole_0,axiom,( ! [A,B] : k2_xboole_0(A,A) = A ), file(xboole_0,k2_xboole_0), []). fof(involutiveness_k1_real_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => k1_real_1(k1_real_1(A)) = A ) ), file(real_1,k1_real_1), []). fof(involutiveness_k3_subset_1,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => k3_subset_1(A,k3_subset_1(A,B)) = B ) ), file(subset_1,k3_subset_1), []). fof(involutiveness_k4_relat_1,axiom,( ! [A] : ( v1_relat_1(A) => k4_relat_1(k4_relat_1(A)) = A ) ), file(relat_1,k4_relat_1), []). fof(involutiveness_k4_xcmplx_0,axiom,( ! [A] : ( v1_xcmplx_0(A) => k4_xcmplx_0(k4_xcmplx_0(A)) = A ) ), file(xcmplx_0,k4_xcmplx_0), []). fof(involutiveness_k6_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => k6_relset_1(A,B,k6_relset_1(A,B,C)) = C ) ), file(relset_1,k6_relset_1), []). fof(rc1_afinsq_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v5_ordinal1(A) & v1_finset_1(A) ) ), file(afinsq_1,rc1_afinsq_1), []). fof(rc1_amistd_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) & ~ v1_realset1(A) ) ), file(amistd_1,rc1_amistd_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_funcop_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_funcop_1(A) ) ), file(funcop_1,rc1_funcop_1), []). fof(rc1_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc1_funct_1), []). fof(rc1_funct_2,axiom,( ! [A,B] : ? [C] : ( m1_relset_1(C,A,B) & v1_relat_1(C) & v1_funct_1(C) & v1_funct_2(C,A,B) ) ), file(funct_2,rc1_funct_2), []). fof(rc1_necklace,axiom,( ? [A] : ( l1_orders_2(A) & ~ v3_struct_0(A) & v1_necklace(A) ) ), file(necklace,rc1_necklace), []). fof(rc1_orders_2,axiom,( ? [A] : ( l1_orders_2(A) & v1_orders_2(A) ) ), file(orders_2,rc1_orders_2), []). fof(rc1_orders_3,axiom,( ? [A] : ( l1_orders_2(A) & ~ v3_struct_0(A) & v1_orders_2(A) & v2_orders_2(A) & v3_orders_2(A) & v4_orders_2(A) & v1_orders_3(A) ) ), file(orders_3,rc1_orders_3), []). 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_tex_2,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_realset1(A) ) ), file(tex_2,rc1_tex_2), []). fof(rc1_xreal_0,axiom,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(xreal_0,rc1_xreal_0), []). fof(rc2_finset_1,axiom,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v2_funct_1(B) & v1_ordinal1(B) & v2_ordinal1(B) & v3_ordinal1(B) & v4_ordinal2(B) & v1_finset_1(B) ) ), file(finset_1,rc2_finset_1), []). fof(rc2_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) ), file(funct_1,rc2_funct_1), []). fof(rc2_funct_2,axiom,( ! [A] : ? [B] : ( m1_relset_1(B,A,A) & v1_relat_1(B) & v1_funct_1(B) & v2_funct_1(B) & v1_funct_2(B,A,A) & v2_funct_2(B,A,A) & v3_funct_2(B,A,A) ) ), file(funct_2,rc2_funct_2), []). fof(rc2_orders_2,axiom,( ? [A] : ( l1_orders_2(A) & ~ v3_struct_0(A) & v1_orders_2(A) & v2_orders_2(A) & v3_orders_2(A) & v4_orders_2(A) ) ), file(orders_2,rc2_orders_2), []). fof(rc2_orders_3,axiom,( ? [A] : ( l1_orders_2(A) & v3_struct_0(A) & v1_orders_2(A) & v2_orders_2(A) & v3_orders_2(A) & v4_orders_2(A) & v1_orders_3(A) ) ), file(orders_3,rc2_orders_3), []). 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_tex_2,axiom,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_tex_2(B,k1_zfmisc_1(A)) ) ), file(tex_2,rc2_tex_2), []). 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_afinsq_1,axiom,( ! [A] : ? [B] : ( m1_relset_1(B,k5_numbers,A) & v1_relat_1(B) & v1_funct_1(B) & v5_ordinal1(B) & v1_finset_1(B) ) ), file(afinsq_1,rc3_afinsq_1), []). 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_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ), file(funct_1,rc3_funct_1), []). fof(rc3_struct_0,axiom,( ? [A] : ( l1_struct_0(A) & ~ v3_struct_0(A) ) ), file(struct_0,rc3_struct_0), []). fof(rc3_tex_2,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_tex_2(B,k1_zfmisc_1(A)) ) ) ), file(tex_2,rc3_tex_2), []). 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_afinsq_1,axiom,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v5_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_finset_1(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(afinsq_1,rc4_afinsq_1), []). 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_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v3_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc4_funct_1), []). fof(rc4_tex_2,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & ~ v1_tex_2(B,k1_zfmisc_1(A)) ) ) ), file(tex_2,rc4_tex_2), []). 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(rc5_tex_2,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & v1_realset1(B) ) ) ), file(tex_2,rc5_tex_2), []). fof(rc6_tex_2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & ~ v1_realset1(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & v1_realset1(B) & v1_tex_2(B,k1_zfmisc_1(A)) ) ) ), file(tex_2,rc6_tex_2), []). fof(rc7_tex_2,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & ~ v1_realset1(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & ~ v1_realset1(B) & ~ v1_tex_2(B,k1_zfmisc_1(A)) ) ) ), file(tex_2,rc7_tex_2), []). 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_real_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => k1_real_1(A) = k4_xcmplx_0(A) ) ), file(real_1,k1_real_1), []). fof(redefinition_k1_relset_1,axiom,( ! [A,B,C,D] : ( ( m1_relset_1(C,A,B) & m1_relset_1(D,A,B) ) => k1_relset_1(A,B,C,D) = k2_xboole_0(C,D) ) ), file(relset_1,k1_relset_1), []). fof(redefinition_k2_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k2_nat_1(A,B) = k3_xcmplx_0(A,B) ) ), file(nat_1,k2_nat_1), []). fof(redefinition_k2_quin_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) & m1_subset_1(C,k1_numbers) ) => k2_quin_1(A,B,C) = k1_quin_1(A,B,C) ) ), file(quin_1,k2_quin_1), []). fof(redefinition_k3_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k3_real_1(A,B) = k2_xcmplx_0(A,B) ) ), file(real_1,k3_real_1), []). fof(redefinition_k4_card_1,axiom,( ! [A] : ( v1_finset_1(A) => k4_card_1(A) = k1_card_1(A) ) ), file(card_1,k4_card_1), []). fof(redefinition_k4_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k4_real_1(A,B) = k3_xcmplx_0(A,B) ) ), file(real_1,k4_real_1), []). fof(redefinition_k5_numbers,axiom,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), []). fof(redefinition_k5_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k5_real_1(A,B) = k6_xcmplx_0(A,B) ) ), file(real_1,k5_real_1), []). fof(redefinition_k6_partfun1,axiom,( ! [A] : k6_partfun1(A) = k6_relat_1(A) ), file(partfun1,k6_partfun1), []). fof(redefinition_k6_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k6_real_1(A,B) = k7_xcmplx_0(A,B) ) ), file(real_1,k6_real_1), []). fof(redefinition_k6_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => k6_relset_1(A,B,C) = k4_relat_1(C) ) ), file(relset_1,k6_relset_1), []). fof(redefinition_k6_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k6_subset_1(A,B,C) = k4_xboole_0(B,C) ) ), file(subset_1,k6_subset_1), []). fof(redefinition_k7_square_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => k7_square_1(A) = k5_square_1(A) ) ), file(square_1,k7_square_1), []). fof(redefinition_k9_square_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => k9_square_1(A) = k8_square_1(A) ) ), file(square_1,k9_square_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(reflexivity_r5_waybel_1,axiom,( ! [A,B] : ( ( l1_orders_2(A) & l1_orders_2(B) ) => r5_waybel_1(A,A) ) ), file(waybel_1,r5_waybel_1), []). 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(spc9_boole,axiom,( ~ v1_xboole_0(9) ), file(boole,spc9_boole), []). fof(spc9_numerals,axiom, ( v2_xreal_0(9) & m1_subset_1(9,k5_numbers) ), file(numerals,spc9_numerals), []). 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(t15_quin_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ~ ( A != 0 & r1_xreal_0(0,k1_quin_1(A,B,C)) & k2_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(A,k5_square_1(D)),k3_xcmplx_0(B,D)),C) = 0 & D != k7_xcmplx_0(k6_xcmplx_0(k4_xcmplx_0(B),k8_square_1(k1_quin_1(A,B,C))),k3_xcmplx_0(2,A)) & D != k7_xcmplx_0(k2_xcmplx_0(k4_xcmplx_0(B),k8_square_1(k1_quin_1(A,B,C))),k3_xcmplx_0(2,A)) ) ) ) ) ) ), file(quin_1,t15_quin_1), []). fof(t18_necklace,axiom,( ! [A] : ( l1_orders_2(A) => ! [B] : ( l1_orders_2(B) => ( r5_waybel_1(A,B) => k1_card_1(u1_orders_2(A)) = k1_card_1(u1_orders_2(B)) ) ) ) ), file(necklace,t18_necklace), []). fof(t19_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ~ ( 0 != A & r1_xreal_0(A,0) ) ) ), file(nat_1,t19_nat_1), []). fof(t19_necklace,axiom,( ! [A] : ( v4_ordinal2(A) => u1_orders_2(k4_necklace(A)) = k2_xboole_0(a_1_1_necklace(A),a_1_3_necklace(A)) ) ), file(necklace,t19_necklace), []). 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_card_5,axiom, ( 1 = k1_tarski(0) & 2 = k2_tarski(0,1) ), file(card_5,t1_card_5), []). 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_yellow11,axiom,( 3 = k1_enumset1(0,1,2) ), file(yellow11,t1_yellow11), []). fof(t20_necklace,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( r2_hidden(B,u1_orders_2(k4_necklace(A))) <=> ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & ~ r1_xreal_0(A,k1_nat_1(C,1)) & ( B = k4_tarski(C,k1_nat_1(C,1)) | B = k4_tarski(k1_nat_1(C,1),C) ) ) ) ) ), file(necklace,t20_necklace), []). fof(t21_necklace,axiom,( ! [A] : ( v4_ordinal2(A) => u1_struct_0(k4_necklace(A)) = A ) ), file(necklace,t21_necklace), []). fof(t25_mcart_1,axiom,( ! [A,B,C,D] : k2_zfmisc_1(k2_tarski(A,B),k2_tarski(C,D)) = k2_enumset1(k4_tarski(A,C),k4_tarski(A,D),k4_tarski(B,C),k4_tarski(B,D)) ), file(mcart_1,t25_mcart_1), []). fof(t28_necklace,axiom,( ! [A] : ( v4_ordinal2(A) => ( ~ r1_xreal_0(A,0) => k1_card_1(u1_orders_2(k4_necklace(A))) = k3_xcmplx_0(2,k6_xcmplx_0(A,1)) ) ) ), file(necklace,t28_necklace), []). fof(t2_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(A,0) = 0 ) ), file(arithm,t2_arithm), []). fof(t2_cqc_the1,axiom,( ! [A] : ( v4_ordinal2(A) => ~ ( r1_xreal_0(A,1) & A != 0 & A != 1 ) ) ), file(cqc_the1,t2_cqc_the1), []). 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(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(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_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(1,A) = A ) ), file(arithm,t3_arithm), []). fof(t3_boole,axiom,( ! [A] : k4_xboole_0(A,k1_xboole_0) = A ), file(boole,t3_boole), []). fof(t3_necklace,axiom,( ! [A,B,C,D,E,F] : k2_zfmisc_1(k1_enumset1(A,B,C),k1_enumset1(D,E,F)) = k1_bvfunc24(k4_tarski(A,D),k4_tarski(A,E),k4_tarski(A,F),k4_tarski(B,D),k4_tarski(B,E),k4_tarski(B,F),k4_tarski(C,D),k4_tarski(C,E),k4_tarski(C,F)) ), file(necklace,t3_necklace), []). 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(t3_xboole_0,axiom,( ! [A,B] : ( ~ ( ~ r1_xboole_0(A,B) & ! [C] : ~ ( r2_hidden(C,A) & r2_hidden(C,B) ) ) & ~ ( ? [C] : ( r2_hidden(C,A) & r2_hidden(C,B) ) & r1_xboole_0(A,B) ) ) ), file(xboole_0,t3_xboole_0), []). fof(t47_card_1,axiom,( k1_card_1(k1_xboole_0) = k1_xboole_0 ), file(card_1,t47_card_1), []). fof(t4_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k6_xcmplx_0(A,0) = A ) ), file(arithm,t4_arithm), []). fof(t4_boole,axiom,( ! [A] : k4_xboole_0(k1_xboole_0,A) = k1_xboole_0 ), file(boole,t4_boole), []). fof(t4_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( r1_xreal_0(A,B) & ~ v2_xreal_0(B) & v2_xreal_0(A) ) ) ) ), file(real,t4_real), []). fof(t4_subset,axiom,( ! [A,B,C] : ( ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) ) => m1_subset_1(A,C) ) ), file(subset,t4_subset), []). fof(t5_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k7_xcmplx_0(0,A) = 0 ) ), file(arithm,t5_arithm), []). fof(t5_cqc_the1,axiom,( ! [A] : ( v4_ordinal2(A) => ~ ( r1_xreal_0(A,4) & A != 0 & A != 1 & A != 2 & A != 3 & A != 4 ) ) ), file(cqc_the1,t5_cqc_the1), []). 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(t63_card_2,axiom,( ! [A] : ( v1_finset_1(A) => ! [B] : ( v1_finset_1(B) => ( r1_tarski(B,A) => k4_card_1(k4_xboole_0(A,B)) = k5_real_1(k4_card_1(A),k4_card_1(B)) ) ) ) ), file(card_2,t63_card_2), []). fof(t65_card_2,axiom,( ! [A] : ( v1_finset_1(A) => ! [B] : ( v1_finset_1(B) => k4_card_1(k2_zfmisc_1(A,B)) = k2_nat_1(k4_card_1(A),k4_card_1(B)) ) ) ), file(card_2,t65_card_2), []). fof(t66_card_1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => A = k1_card_1(A) ) ), file(card_1,t66_card_1), []). fof(t6_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k7_xcmplx_0(A,1) = A ) ), file(arithm,t6_arithm), []). fof(t6_boole,axiom,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), []). fof(t6_necklace,axiom,( ! [A] : k1_card_1(k6_partfun1(A)) = k1_card_1(A) ), file(necklace,t6_necklace), []). 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(t86_xboole_1,axiom,( ! [A,B,C] : ( ( r1_tarski(A,B) & r1_xboole_0(A,C) ) => r1_tarski(A,k4_xboole_0(B,C)) ) ), file(xboole_1,t86_xboole_1), []). fof(t89_square_1,axiom,( ! [A] : ( v1_xreal_0(A) => ( r1_xreal_0(0,A) => k8_square_1(k5_square_1(A)) = A ) ) ), file(square_1,t89_square_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(t9_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ( ( r1_xreal_0(A,B) & r1_xreal_0(C,D) ) => r1_xreal_0(k2_xcmplx_0(A,C),k2_xcmplx_0(B,D)) ) ) ) ) ) ), file(xreal_1,t9_xreal_1), []).