% Mizar problem: t40_groeb_1,groeb_1,4321,10 fof(t40_groeb_1,conjecture,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & v2_bagorder(B,A) & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) ) => ! [C] : ( ( ~ v3_struct_0(C) & v2_group_1(C) & v4_group_1(C) & v7_group_1(C) & v3_rlvect_1(C) & v4_rlvect_1(C) & v5_rlvect_1(C) & v6_rlvect_1(C) & v7_vectsp_1(C) & v9_vectsp_1(C) & ~ v10_vectsp_1(C) & l3_vectsp_1(C) ) => ! [D] : ( ( ~ v1_xboole_0(D) & v1_ideal_1(D,k30_polynom1(A,C)) & v2_ideal_1(D,k30_polynom1(A,C)) & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) ) => ! [E] : ( ( ~ v1_xboole_0(E) & v1_finset_1(E) & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) ) => ! [F] : ( ( ~ v1_xboole_0(F) & v1_finset_1(F) & m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) ) => ( ( r2_groeb_1(A,B,C,E,D) & r4_groeb_1(A,B,C,E) & r2_groeb_1(A,B,C,F,D) & r4_groeb_1(A,B,C,F) ) => E = F ) ) ) ) ) ) ) ), inference(mizar_bg_added,[status(thm)],[abstractness_v3_vectsp_1,antisymmetry_r2_hidden,cc10_rewrite1,cc11_rewrite1,cc12_rewrite1,cc13_rewrite1,cc14_rewrite1,cc15_rewrite1,cc1_algstr_1,cc1_binom,cc1_card_1,cc1_finset_1,cc1_gcd_1,cc1_ideal_1,cc1_nat_1,cc1_ordinal1,cc1_polynom4,cc1_polynom7,cc1_relat_1,cc1_relset_1,cc1_termord,cc1_vectsp_1,cc2_algstr_1,cc2_binom,cc2_card_1,cc2_finset_1,cc2_gcd_1,cc2_ideal_1,cc2_int_1,cc2_nat_1,cc2_ordinal1,cc2_polynom1,cc2_rewrite1,cc2_vectsp_1,cc3_algstr_1,cc3_binom,cc3_card_1,cc3_gcd_1,cc3_int_1,cc3_nat_1,cc3_ordinal1,cc3_polynom1,cc3_polynom7,cc3_rewrite1,cc3_vectsp_1,cc4_algstr_1,cc4_binom,cc4_gcd_1,cc4_int_1,cc4_polynom1,cc4_polynom7,cc4_rewrite1,cc4_vectsp_1,cc5_binom,cc5_gcd_1,cc5_polynom1,cc5_rewrite1,cc5_vectsp_1,cc6_polynom1,cc6_rewrite1,cc7_rewrite1,cc8_rewrite1,cc9_rewrite1,commutativity_k13_ideal_1,commutativity_k23_polynom1,commutativity_k2_xboole_0,commutativity_k3_xboole_0,commutativity_k4_rlvect_1,commutativity_k4_subset_1,d15_ideal_1,d1_groeb_1,d1_tarski,d21_polynom1,d21_vectsp_1,d22_polynom1,d23_polynom1,d27_polynom1,d2_polynom7,d2_xboole_0,d3_xboole_0,d4_groeb_1,d4_xboole_0,d6_groeb_1,d6_termord,d7_groeb_1,d7_polyred,d7_termord,d8_polyred,d9_polynom1,d9_polyred,dt_g3_vectsp_1,dt_k12_polynom1,dt_k13_ideal_1,dt_k13_polynom1,dt_k14_polynom1,dt_k15_polynom1,dt_k16_polynom1,dt_k1_funct_1,dt_k1_groeb_1,dt_k1_group_1,dt_k1_numbers,dt_k1_rlvect_1,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k22_polynom1,dt_k23_polynom1,dt_k24_polynom1,dt_k25_polynom1,dt_k26_polynom1,dt_k27_polynom1,dt_k28_polynom1,dt_k2_groeb_1,dt_k2_group_1,dt_k2_rlvect_1,dt_k2_termord,dt_k2_vectsp_1,dt_k2_xboole_0,dt_k2_zfmisc_1,dt_k30_polynom1,dt_k3_polyred,dt_k3_termord,dt_k3_xboole_0,dt_k4_rlvect_1,dt_k4_subset_1,dt_k4_termord,dt_k4_xboole_0,dt_k5_numbers,dt_k5_ordinal2,dt_k5_rlvect_1,dt_k6_rlvect_1,dt_k6_subset_1,dt_k6_termord,dt_k7_ideal_1,dt_k8_funct_2,dt_l1_group_1,dt_l1_rlvect_1,dt_l1_struct_0,dt_l1_vectsp_1,dt_l2_struct_0,dt_l2_vectsp_1,dt_l3_vectsp_1,dt_m1_pboole,dt_m1_polynom1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_group_1,dt_u1_rlvect_1,dt_u1_struct_0,dt_u1_vectsp_1,dt_u2_struct_0,existence_l1_group_1,existence_l1_rlvect_1,existence_l1_struct_0,existence_l1_vectsp_1,existence_l2_struct_0,existence_l2_vectsp_1,existence_l3_vectsp_1,existence_m1_pboole,existence_m1_polynom1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,fc10_finset_1,fc11_finset_1,fc12_finset_1,fc12_relat_1,fc14_finset_1,fc14_polynom1,fc15_ideal_1,fc16_ideal_1,fc17_ideal_1,fc18_ideal_1,fc1_finset_1,fc1_ideal_1,fc1_ordinal2,fc1_polynom1,fc1_polynom2,fc1_polynom7,fc1_polyred,fc1_relat_1,fc1_struct_0,fc1_subset_1,fc1_termord,fc20_polynom1,fc25_polynom1,fc2_ideal_1,fc2_ordinal1,fc2_polynom2,fc2_polynom7,fc2_relat_1,fc2_subset_1,fc2_termord,fc31_ideal_1,fc32_polynom1,fc33_polynom1,fc34_polynom1,fc35_polynom1,fc36_polynom1,fc37_polynom1,fc38_polynom1,fc39_polynom1,fc3_ideal_1,fc3_polynom1,fc3_polynom2,fc3_relat_1,fc40_polynom1,fc41_polynom1,fc42_polynom1,fc43_polynom1,fc44_polynom1,fc4_relat_1,fc4_subset_1,fc4_termord,fc5_polynom1,fc5_polyred,fc9_finset_1,fraenkel_a_4_0_groeb_1,fraenkel_a_4_11_groeb_1,fraenkel_a_4_12_groeb_1,fraenkel_a_4_8_groeb_1,free_g3_vectsp_1,idempotence_k13_ideal_1,idempotence_k2_xboole_0,idempotence_k3_xboole_0,idempotence_k4_subset_1,involutiveness_k24_polynom1,l4_groeb_1,rc10_polynom1,rc10_vectsp_1,rc11_polynom1,rc12_polynom1,rc13_polynom1,rc14_ideal_1,rc14_polynom1,rc1_binom,rc1_card_1,rc1_finset_1,rc1_gcd_1,rc1_ideal_1,rc1_int_1,rc1_nat_1,rc1_ordinal1,rc1_polynom2,rc1_polynom4,rc1_polynom7,rc1_polyred,rc1_relat_1,rc1_rewrite1,rc1_subset_1,rc1_termord,rc2_binom,rc2_card_1,rc2_finset_1,rc2_gcd_1,rc2_ideal_1,rc2_int_1,rc2_nat_1,rc2_ordinal1,rc2_polynom1,rc2_polynom7,rc2_polyred,rc2_relat_1,rc2_rewrite1,rc2_subset_1,rc2_termord,rc3_binom,rc3_finset_1,rc3_gcd_1,rc3_ideal_1,rc3_nat_1,rc3_ordinal1,rc3_polynom7,rc3_relat_1,rc3_rewrite1,rc3_struct_0,rc3_termord,rc4_binom,rc4_finset_1,rc4_gcd_1,rc4_ideal_1,rc4_polynom7,rc4_rewrite1,rc4_struct_0,rc5_gcd_1,rc5_ideal_1,rc5_polynom7,rc5_struct_0,rc6_ideal_1,rc6_polynom1,rc6_vectsp_1,rc7_ideal_1,rc7_polynom1,rc7_polynom7,rc7_vectsp_1,rc8_ideal_1,rc8_polynom1,rc8_polynom7,rc8_vectsp_1,rc9_polynom1,rc9_vectsp_1,redefinition_k13_ideal_1,redefinition_k14_polynom1,redefinition_k15_polynom1,redefinition_k1_groeb_1,redefinition_k23_polynom1,redefinition_k4_rlvect_1,redefinition_k4_subset_1,redefinition_k5_numbers,redefinition_k6_subset_1,redefinition_k8_funct_2,redefinition_m1_polynom1,redefinition_m2_relset_1,redefinition_m2_subset_1,redefinition_r6_pboole,reflexivity_r1_tarski,reflexivity_r3_polynom1,reflexivity_r6_pboole,symmetry_r6_pboole,t10_termord,t12_groeb_1,t12_termord,t13_groeb_1,t14_groeb_1,t15_groeb_1,t15_ideal_1,t16_groeb_1,t16_rlvect_1,t17_groeb_1,t17_termord,t18_groeb_1,t1_boole,t1_polynom7,t1_subset,t29_polyred,t2_boole,t2_polyred,t2_subset,t2_tarski,t31_polyred,t32_polyred,t36_polyred,t37_xboole_1,t3_boole,t3_polyred,t3_subset,t4_boole,t4_subset,t5_groeb_1,t5_subset,t5_termord,t6_boole,t79_polynom1,t7_boole,t7_groeb_1,t7_termord,t80_polynom1,t82_polynom1,t8_boole,t9_groeb_1]), [file(groeb_1,t40_groeb_1)]). fof(abstractness_v3_vectsp_1,axiom,( ! [A] : ( l3_vectsp_1(A) => ( v3_vectsp_1(A) => A = g3_vectsp_1(u1_struct_0(A),u1_rlvect_1(A),u1_group_1(A),u1_vectsp_1(A),u2_struct_0(A)) ) ) ), file(vectsp_1,v3_vectsp_1), []). fof(antisymmetry_r2_hidden,axiom,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), []). fof(cc10_rewrite1,axiom,( ! [A] : ( ( v1_relat_1(A) & v5_rewrite1(A) ) => ( v1_relat_1(A) & v4_rewrite1(A) ) ) ), file(rewrite1,cc10_rewrite1), []). fof(cc11_rewrite1,axiom,( ! [A] : ( ( v1_relat_1(A) & v2_rewrite1(A) & v4_rewrite1(A) ) => ( v1_relat_1(A) & v8_rewrite1(A) ) ) ), file(rewrite1,cc11_rewrite1), []). fof(cc12_rewrite1,axiom,( ! [A] : ( ( v1_xboole_0(A) & v1_relat_1(A) ) => ( v1_relat_1(A) & v4_rewrite1(A) & v5_rewrite1(A) & v6_rewrite1(A) & v7_rewrite1(A) & v8_rewrite1(A) & v9_rewrite1(A) ) ) ), file(rewrite1,cc12_rewrite1), []). fof(cc13_rewrite1,axiom,( ! [A] : ( ( v1_relat_1(A) & v3_rewrite1(A) & v9_rewrite1(A) ) => ( v1_relat_1(A) & v4_rewrite1(A) & v5_rewrite1(A) & v7_rewrite1(A) & v8_rewrite1(A) & v9_rewrite1(A) ) ) ), file(rewrite1,cc13_rewrite1), []). fof(cc14_rewrite1,axiom,( ! [A] : ( ( v1_relat_1(A) & v10_rewrite1(A) ) => ( v1_relat_1(A) & v2_relat_2(A) & v1_rewrite1(A) & v2_rewrite1(A) & v3_rewrite1(A) & v4_rewrite1(A) & v5_rewrite1(A) & v7_rewrite1(A) & v8_rewrite1(A) & v9_rewrite1(A) ) ) ), file(rewrite1,cc14_rewrite1), []). fof(cc15_rewrite1,axiom,( ! [A] : ( ( v1_relat_1(A) & v3_rewrite1(A) & v7_rewrite1(A) ) => ( v1_relat_1(A) & v10_rewrite1(A) ) ) ), file(rewrite1,cc15_rewrite1), []). fof(cc1_algstr_1,axiom,( ! [A] : ( l1_rlvect_1(A) => ( ( ~ v3_struct_0(A) & v6_algstr_1(A) ) => ( ~ v3_struct_0(A) & v2_algstr_1(A) & v3_algstr_1(A) & v4_algstr_1(A) & v5_algstr_1(A) ) ) ) ), file(algstr_1,cc1_algstr_1), []). fof(cc1_binom,axiom,( ! [A] : ( l1_rlvect_1(A) => ( ( ~ v3_struct_0(A) & v2_algstr_1(A) & v3_algstr_1(A) ) => ( ~ v3_struct_0(A) & v1_binom(A) ) ) ) ), file(binom,cc1_binom), []). fof(cc1_card_1,axiom,( ! [A] : ( v1_card_1(A) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) ) ) ), file(card_1,cc1_card_1), []). fof(cc1_finset_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_finset_1(A) ) ), file(finset_1,cc1_finset_1), []). fof(cc1_gcd_1,axiom,( ! [A] : ( l1_vectsp_1(A) => ( ( ~ v3_struct_0(A) & v7_group_1(A) & v6_vectsp_1(A) ) => ( ~ v3_struct_0(A) & v8_vectsp_1(A) ) ) ) ), file(gcd_1,cc1_gcd_1), []). fof(cc1_ideal_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v7_group_1(A) & l1_group_1(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( ( ~ v1_xboole_0(B) & v2_ideal_1(B,A) ) => ( ~ v1_xboole_0(B) & v3_ideal_1(B,A) ) ) ) ) ), file(ideal_1,cc1_ideal_1), []). fof(cc1_nat_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(nat_1,cc1_nat_1), []). fof(cc1_ordinal1,axiom,( ! [A] : ( v3_ordinal1(A) => ( v1_ordinal1(A) & v2_ordinal1(A) ) ) ), file(ordinal1,cc1_ordinal1), []). fof(cc1_polynom4,axiom,( ! [A] : ( l3_vectsp_1(A) => ( ( ~ v3_struct_0(A) & v4_group_1(A) & v7_group_1(A) & v4_vectsp_1(A) & v8_vectsp_1(A) & v9_vectsp_1(A) & v1_algstr_1(A) & v3_algstr_1(A) ) => ( ~ v3_struct_0(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v4_vectsp_1(A) & v6_vectsp_1(A) & v8_vectsp_1(A) & v2_vectsp_2(A) & v1_algstr_1(A) & v3_algstr_1(A) ) ) ) ), file(polynom4,cc1_polynom4), []). fof(cc1_polynom7,axiom,( ! [A] : ( l2_struct_0(A) => ( ~ v3_realset2(A) => ~ v3_struct_0(A) ) ) ), file(polynom7,cc1_polynom7), []). fof(cc1_relat_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_relat_1(A) ) ), file(relat_1,cc1_relat_1), []). fof(cc1_relset_1,axiom,( ! [A,B,C] : ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) => v1_relat_1(C) ) ), file(relset_1,cc1_relset_1), []). fof(cc1_termord,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ! [B] : ( m1_relset_1(B,k14_polynom1(A),k14_polynom1(A)) => ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_2(B) & v4_relat_2(B) & v8_relat_2(B) & v2_bagorder(B,A) ) => ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_2(B) & v4_relat_2(B) & v8_relat_2(B) & v2_bagorder(B,A) & v1_wellord1(B) ) ) ) ) ), file(termord,cc1_termord), []). fof(cc1_vectsp_1,axiom,( ! [A] : ( l3_vectsp_1(A) => ( ( ~ v3_struct_0(A) & v7_vectsp_1(A) ) => ( ~ v3_struct_0(A) & v4_vectsp_1(A) & v5_vectsp_1(A) ) ) ) ), file(vectsp_1,cc1_vectsp_1), []). fof(cc2_algstr_1,axiom,( ! [A] : ( l1_rlvect_1(A) => ( ( ~ v3_struct_0(A) & v2_algstr_1(A) & v3_algstr_1(A) & v4_algstr_1(A) & v5_algstr_1(A) ) => ( ~ v3_struct_0(A) & v6_algstr_1(A) ) ) ) ), file(algstr_1,cc2_algstr_1), []). fof(cc2_binom,axiom,( ! [A] : ( l1_rlvect_1(A) => ( ( ~ v3_struct_0(A) & v1_binom(A) ) => ( ~ v3_struct_0(A) & v2_algstr_1(A) & v3_algstr_1(A) ) ) ) ), file(binom,cc2_binom), []). fof(cc2_card_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_card_1(A) ) ) ), file(card_1,cc2_card_1), []). 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_gcd_1,axiom,( ! [A] : ( l3_vectsp_1(A) => ( ( ~ v3_struct_0(A) & v7_group_1(A) & v4_vectsp_1(A) ) => ( ~ v3_struct_0(A) & v4_vectsp_1(A) & v5_vectsp_1(A) & v7_vectsp_1(A) ) ) ) ), file(gcd_1,cc2_gcd_1), []). fof(cc2_ideal_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v7_group_1(A) & l1_group_1(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( ( ~ v1_xboole_0(B) & v3_ideal_1(B,A) ) => ( ~ v1_xboole_0(B) & v2_ideal_1(B,A) ) ) ) ) ), file(ideal_1,cc2_ideal_1), []). fof(cc2_int_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_int_1(A) ) ) ), file(int_1,cc2_int_1), []). fof(cc2_nat_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc2_nat_1), []). fof(cc2_ordinal1,axiom,( ! [A] : ( ( v1_ordinal1(A) & v2_ordinal1(A) ) => v3_ordinal1(A) ) ), file(ordinal1,cc2_ordinal1), []). fof(cc2_polynom1,axiom,( ! [A] : ( l2_vectsp_1(A) => ( ( ~ v3_struct_0(A) & ~ v10_vectsp_1(A) ) => ( ~ v3_struct_0(A) & ~ v3_realset2(A) ) ) ) ), file(polynom1,cc2_polynom1), []). fof(cc2_rewrite1,axiom,( ! [A] : ( ( v1_relat_1(A) & v3_rewrite1(A) ) => ( v1_relat_1(A) & v2_relat_2(A) & v1_rewrite1(A) ) ) ), file(rewrite1,cc2_rewrite1), []). fof(cc2_vectsp_1,axiom,( ! [A] : ( l3_vectsp_1(A) => ( ( ~ v3_struct_0(A) & v4_vectsp_1(A) & v5_vectsp_1(A) ) => ( ~ v3_struct_0(A) & v7_vectsp_1(A) ) ) ) ), file(vectsp_1,cc2_vectsp_1), []). fof(cc3_algstr_1,axiom,( ! [A] : ( l1_rlvect_1(A) => ( ( ~ v3_struct_0(A) & v6_algstr_1(A) ) => ( ~ v3_struct_0(A) & v6_rlvect_1(A) ) ) ) ), file(algstr_1,cc3_algstr_1), []). fof(cc3_binom,axiom,( ! [A] : ( l1_rlvect_1(A) => ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v3_algstr_1(A) ) => ( ~ v3_struct_0(A) & v2_algstr_1(A) ) ) ) ), file(binom,cc3_binom), []). fof(cc3_card_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_finset_1(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_card_1(A) ) ) ), file(card_1,cc3_card_1), []). fof(cc3_gcd_1,axiom,( ! [A] : ( l3_vectsp_1(A) => ( ( ~ v3_struct_0(A) & v7_group_1(A) & v5_vectsp_1(A) ) => ( ~ v3_struct_0(A) & v4_vectsp_1(A) & v5_vectsp_1(A) & v7_vectsp_1(A) ) ) ) ), file(gcd_1,cc3_gcd_1), []). fof(cc3_int_1,axiom,( ! [A] : ( v4_ordinal2(A) => v1_int_1(A) ) ), file(int_1,cc3_int_1), []). fof(cc3_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ( v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc3_nat_1), []). fof(cc3_ordinal1,axiom,( ! [A] : ( v1_xboole_0(A) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) ) ) ), file(ordinal1,cc3_ordinal1), []). fof(cc3_polynom1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v7_seqm_3(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) ) ) ), file(polynom1,cc3_polynom1), []). fof(cc3_polynom7,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & l2_struct_0(B) ) => ! [C] : ( m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) => ( ( v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & v3_polynom7(C,A,B) ) => ( v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(C,k14_polynom1(A),B) ) ) ) ) ), file(polynom7,cc3_polynom7), []). fof(cc3_rewrite1,axiom,( ! [A] : ( ( v1_relat_1(A) & v2_relat_2(A) & v1_rewrite1(A) ) => ( v1_relat_1(A) & v3_rewrite1(A) ) ) ), file(rewrite1,cc3_rewrite1), []). fof(cc3_vectsp_1,axiom,( ! [A] : ( l1_vectsp_1(A) => ( ( ~ v3_struct_0(A) & v2_group_1(A) ) => ( ~ v3_struct_0(A) & v6_vectsp_1(A) & v8_vectsp_1(A) ) ) ) ), file(vectsp_1,cc3_vectsp_1), []). fof(cc4_algstr_1,axiom,( ! [A] : ( l1_rlvect_1(A) => ( ( ~ v3_struct_0(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) ) => ( ~ v3_struct_0(A) & v1_algstr_1(A) & v2_algstr_1(A) & v3_algstr_1(A) & v4_algstr_1(A) & v5_algstr_1(A) & v6_algstr_1(A) ) ) ) ), file(algstr_1,cc4_algstr_1), []). fof(cc4_binom,axiom,( ! [A] : ( l1_rlvect_1(A) => ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v2_algstr_1(A) ) => ( ~ v3_struct_0(A) & v3_algstr_1(A) ) ) ) ), file(binom,cc4_binom), []). fof(cc4_gcd_1,axiom,( ! [A] : ( l3_vectsp_1(A) => ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_group_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) ) => ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v2_group_1(A) & v4_group_1(A) & v4_vectsp_1(A) & v5_vectsp_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) ) ) ) ), file(gcd_1,cc4_gcd_1), []). fof(cc4_int_1,axiom,( ! [A] : ( v1_int_1(A) => ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(int_1,cc4_int_1), []). fof(cc4_polynom1,axiom,( ! [A,B] : ( m1_relset_1(B,A,k5_numbers) => ( ( v1_funct_1(B) & v1_funct_2(B,A,k5_numbers) ) => ( v1_funct_1(B) & v1_funct_2(B,A,k5_numbers) & v7_seqm_3(B) ) ) ) ), file(polynom1,cc4_polynom1), []). fof(cc4_polynom7,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & l2_struct_0(B) ) => ! [C] : ( m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) => ( ( v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & v4_polynom7(C,A,B) ) => ( v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(C,k14_polynom1(A),B) & v3_polynom7(C,A,B) ) ) ) ) ), file(polynom7,cc4_polynom7), []). fof(cc4_rewrite1,axiom,( ! [A] : ( ( v1_xboole_0(A) & v1_relat_1(A) ) => ( v1_relat_1(A) & v2_relat_2(A) & v1_rewrite1(A) & v2_rewrite1(A) & v3_rewrite1(A) ) ) ), file(rewrite1,cc4_rewrite1), []). fof(cc4_vectsp_1,axiom,( ! [A] : ( l1_vectsp_1(A) => ( ( ~ v3_struct_0(A) & v6_vectsp_1(A) & v8_vectsp_1(A) ) => ( ~ v3_struct_0(A) & v2_group_1(A) ) ) ) ), file(vectsp_1,cc4_vectsp_1), []). fof(cc5_binom,axiom,( ! [A] : ( l1_rlvect_1(A) => ( ( ~ v3_struct_0(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) ) => ( ~ v3_struct_0(A) & v3_algstr_1(A) ) ) ) ), file(binom,cc5_binom), []). fof(cc5_gcd_1,axiom,( ! [A] : ( l3_vectsp_1(A) => ( ( ~ v3_struct_0(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_group_1(A) & v4_vectsp_1(A) & v5_vectsp_1(A) & v6_vectsp_1(A) & v8_vectsp_1(A) & v9_vectsp_1(A) ) => ( ~ v3_struct_0(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v2_group_1(A) & v7_group_1(A) & v4_vectsp_1(A) & v5_vectsp_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v9_vectsp_1(A) & v3_gcd_1(A) ) ) ) ), file(gcd_1,cc5_gcd_1), []). fof(cc5_polynom1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v1_polynom1(A) ) ) ), file(polynom1,cc5_polynom1), []). fof(cc5_rewrite1,axiom,( ! [A] : ( ( v1_relat_1(A) & v3_rewrite1(A) ) => ( v1_relat_1(A) & v2_rewrite1(A) ) ) ), file(rewrite1,cc5_rewrite1), []). fof(cc5_vectsp_1,axiom,( ! [A] : ( l1_vectsp_1(A) => ( ( ~ v3_struct_0(A) & v7_group_1(A) & v8_vectsp_1(A) ) => ( ~ v3_struct_0(A) & v6_vectsp_1(A) ) ) ) ), file(vectsp_1,cc5_vectsp_1), []). fof(cc6_polynom1,axiom,( ! [A] : ( v1_finset_1(A) => ! [B] : ( m1_pboole(B,A) => v1_polynom1(B) ) ) ), file(polynom1,cc6_polynom1), []). fof(cc6_rewrite1,axiom,( ! [A] : ( ( v1_relat_1(A) & v8_rewrite1(A) ) => ( v1_relat_1(A) & v7_rewrite1(A) ) ) ), file(rewrite1,cc6_rewrite1), []). fof(cc7_rewrite1,axiom,( ! [A] : ( ( v1_relat_1(A) & v7_rewrite1(A) ) => ( v1_relat_1(A) & v8_rewrite1(A) & v9_rewrite1(A) ) ) ), file(rewrite1,cc7_rewrite1), []). fof(cc8_rewrite1,axiom,( ! [A] : ( ( v1_relat_1(A) & v6_rewrite1(A) ) => ( v1_relat_1(A) & v7_rewrite1(A) ) ) ), file(rewrite1,cc8_rewrite1), []). fof(cc9_rewrite1,axiom,( ! [A] : ( ( v1_relat_1(A) & v8_rewrite1(A) ) => ( v1_relat_1(A) & v5_rewrite1(A) ) ) ), file(rewrite1,cc9_rewrite1), []). fof(commutativity_k13_ideal_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) ) => k13_ideal_1(A,B,C) = k13_ideal_1(A,C,B) ) ), file(ideal_1,k13_ideal_1), []). fof(commutativity_k23_polynom1,axiom,( ! [A,B,C,D] : ( ( ~ v3_struct_0(B) & v3_rlvect_1(B) & v5_rlvect_1(B) & l1_rlvect_1(B) & v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) & v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(B)) & m1_relset_1(D,k14_polynom1(A),u1_struct_0(B)) ) => k23_polynom1(A,B,C,D) = k23_polynom1(A,B,D,C) ) ), file(polynom1,k23_polynom1), []). 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_k3_xboole_0,axiom,( ! [A,B] : k3_xboole_0(A,B) = k3_xboole_0(B,A) ), file(xboole_0,k3_xboole_0), []). fof(commutativity_k4_rlvect_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & l1_rlvect_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => k4_rlvect_1(A,B,C) = k4_rlvect_1(A,C,B) ) ), file(rlvect_1,k4_rlvect_1), []). fof(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(d15_ideal_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l3_vectsp_1(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( ~ v1_xboole_0(B) => ! [C] : ( ( ~ v1_xboole_0(C) & v1_ideal_1(C,A) & v2_ideal_1(C,A) & v3_ideal_1(C,A) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) ) => ( C = k7_ideal_1(A,B) <=> ( r1_tarski(B,C) & ! [D] : ( ( ~ v1_xboole_0(D) & v1_ideal_1(D,A) & v2_ideal_1(D,A) & v3_ideal_1(D,A) & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) ) => ( r1_tarski(B,D) => r1_tarski(C,D) ) ) ) ) ) ) ) ) ), file(ideal_1,d15_ideal_1), []). fof(d1_groeb_1,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) ) => ! [C] : ( ( ~ v3_struct_0(C) & v2_group_1(C) & v4_rlvect_1(C) & v5_rlvect_1(C) & v6_rlvect_1(C) & v7_vectsp_1(C) & ~ v3_realset2(C) & l3_vectsp_1(C) ) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) => k2_groeb_1(A,B,C,D) = a_4_0_groeb_1(A,B,C,D) ) ) ) ) ), file(groeb_1,d1_groeb_1), []). fof(d1_tarski,axiom,( ! [A,B] : ( B = k1_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> C = A ) ) ), file(tarski,d1_tarski), []). fof(d21_polynom1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & l1_rlvect_1(B) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(B)) & m2_relset_1(D,k14_polynom1(A),u1_struct_0(B)) ) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,k14_polynom1(A),u1_struct_0(B)) & m2_relset_1(E,k14_polynom1(A),u1_struct_0(B)) ) => ( E = k22_polynom1(A,B,C,D) <=> ! [F] : ( ( v7_seqm_3(F) & v1_polynom1(F) & m1_pboole(F,A) ) => k15_polynom1(A,B,E,F) = k2_rlvect_1(B,k15_polynom1(A,B,C,F),k15_polynom1(A,B,D,F)) ) ) ) ) ) ) ), file(polynom1,d21_polynom1), []). fof(d21_vectsp_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l2_vectsp_1(A) ) => ( v10_vectsp_1(A) <=> k1_rlvect_1(A) = k2_group_1(A) ) ) ), file(vectsp_1,d21_vectsp_1), []). fof(d22_polynom1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & l1_rlvect_1(B) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(B)) & m2_relset_1(D,k14_polynom1(A),u1_struct_0(B)) ) => ( D = k24_polynom1(A,B,C) <=> ! [E] : ( ( v7_seqm_3(E) & v1_polynom1(E) & m1_pboole(E,A) ) => k15_polynom1(A,B,D,E) = k5_rlvect_1(B,k15_polynom1(A,B,C,E)) ) ) ) ) ) ), file(polynom1,d22_polynom1), []). fof(d23_polynom1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & l1_rlvect_1(B) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(B)) & m2_relset_1(D,k14_polynom1(A),u1_struct_0(B)) ) => k25_polynom1(A,B,C,D) = k22_polynom1(A,B,C,k24_polynom1(A,B,D)) ) ) ) ), file(polynom1,d23_polynom1), []). fof(d27_polynom1,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( ( ~ v3_struct_0(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v2_group_1(B) & v7_vectsp_1(B) & ~ v3_realset2(B) & l3_vectsp_1(B) ) => ! [C] : ( ( ~ v3_struct_0(C) & v3_vectsp_1(C) & l3_vectsp_1(C) ) => ( C = k30_polynom1(A,B) <=> ( ! [D] : ( r2_hidden(D,u1_struct_0(C)) <=> ( v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(D,k14_polynom1(A),B) & m2_relset_1(D,k14_polynom1(A),u1_struct_0(B)) ) ) & ! [D] : ( m1_subset_1(D,u1_struct_0(C)) => ! [E] : ( m1_subset_1(E,u1_struct_0(C)) => ! [F] : ( ( v1_funct_1(F) & v1_funct_2(F,k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(F,k14_polynom1(A),B) & m2_relset_1(F,k14_polynom1(A),u1_struct_0(B)) ) => ! [G] : ( ( v1_funct_1(G) & v1_funct_2(G,k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(G,k14_polynom1(A),B) & m2_relset_1(G,k14_polynom1(A),u1_struct_0(B)) ) => ( ( D = F & E = G ) => k2_rlvect_1(C,D,E) = k22_polynom1(A,B,F,G) ) ) ) ) ) & ! [D] : ( m1_subset_1(D,u1_struct_0(C)) => ! [E] : ( m1_subset_1(E,u1_struct_0(C)) => ! [F] : ( ( v1_funct_1(F) & v1_funct_2(F,k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(F,k14_polynom1(A),B) & m2_relset_1(F,k14_polynom1(A),u1_struct_0(B)) ) => ! [G] : ( ( v1_funct_1(G) & v1_funct_2(G,k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(G,k14_polynom1(A),B) & m2_relset_1(G,k14_polynom1(A),u1_struct_0(B)) ) => ( ( D = F & E = G ) => k1_group_1(C,D,E) = k28_polynom1(A,B,F,G) ) ) ) ) ) & k1_rlvect_1(C) = k26_polynom1(A,B) & k2_vectsp_1(C) = k27_polynom1(A,B) ) ) ) ) ) ), file(polynom1,d27_polynom1), []). fof(d2_polynom7,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & l2_struct_0(B) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) ) => ( v1_polynom7(C,A,B) <=> C != k26_polynom1(A,B) ) ) ) ), file(polynom7,d2_polynom7), []). 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_xboole_0,axiom,( ! [A,B,C] : ( C = k3_xboole_0(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( r2_hidden(D,A) & r2_hidden(D,B) ) ) ) ), file(xboole_0,d3_xboole_0), []). fof(d4_groeb_1,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) ) => ! [C] : ( ( v2_group_1(C) & v4_group_1(C) & v7_group_1(C) & v4_rlvect_1(C) & v5_rlvect_1(C) & v6_rlvect_1(C) & v7_vectsp_1(C) & v9_vectsp_1(C) & ~ v3_realset2(C) & l3_vectsp_1(C) ) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) => ! [E] : ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) => ( r2_groeb_1(A,B,C,D,E) <=> ( k7_ideal_1(k30_polynom1(A,C),D) = E & v9_rewrite1(k3_polyred(A,B,C,D)) ) ) ) ) ) ) ) ), file(groeb_1,d4_groeb_1), []). 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(d6_groeb_1,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) ) => ! [C] : ( ( ~ v3_struct_0(C) & l2_vectsp_1(C) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(D,k14_polynom1(A),C) & m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) ) => ( r3_groeb_1(A,B,C,D) <=> k4_termord(A,B,C,D) = k2_group_1(C) ) ) ) ) ) ), file(groeb_1,d6_groeb_1), []). fof(d6_termord,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) ) => ! [C] : ( ( ~ v3_struct_0(C) & l2_struct_0(C) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(D,k14_polynom1(A),C) & m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) ) => ! [E] : ( m1_polynom1(E,A,k14_polynom1(A)) => ( E = k3_termord(A,B,C,D) <=> ( ( k12_polynom1(k14_polynom1(A),C,D) = k1_xboole_0 & r6_pboole(A,E,k16_polynom1(A)) ) | ( r2_hidden(E,k12_polynom1(k14_polynom1(A),C,D)) & ! [F] : ( ( v7_seqm_3(F) & v1_polynom1(F) & m1_pboole(F,A) ) => ( r2_hidden(F,k12_polynom1(k14_polynom1(A),C,D)) => r1_termord(A,B,F,E) ) ) ) ) ) ) ) ) ) ) ), file(termord,d6_termord), []). fof(d7_groeb_1,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) ) => ! [C] : ( ( ~ v3_struct_0(C) & v2_group_1(C) & v4_group_1(C) & v7_group_1(C) & v4_rlvect_1(C) & v5_rlvect_1(C) & v6_rlvect_1(C) & v7_vectsp_1(C) & v9_vectsp_1(C) & ~ v3_realset2(C) & l3_vectsp_1(C) ) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) => ( r4_groeb_1(A,B,C,D) <=> ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(E,k14_polynom1(A),C) & m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) ) => ( r2_hidden(E,D) => ( r3_groeb_1(A,B,C,E) & ~ r7_polyred(A,B,C,E,k6_subset_1(u1_struct_0(k30_polynom1(A,C)),D,k1_groeb_1(A,C,E))) ) ) ) ) ) ) ) ) ), file(groeb_1,d7_groeb_1), []). fof(d7_polyred,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( ( v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) ) => ! [C] : ( ( v2_group_1(C) & v4_group_1(C) & v7_group_1(C) & v4_rlvect_1(C) & v5_rlvect_1(C) & v6_rlvect_1(C) & v7_vectsp_1(C) & v9_vectsp_1(C) & ~ v3_realset2(C) & l3_vectsp_1(C) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(D,k14_polynom1(A),C) & m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) ) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(E,k14_polynom1(A),C) & m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) ) => ! [F] : ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) => ( r5_polyred(A,B,C,D,E,F) <=> ? [G] : ( v1_funct_1(G) & v1_funct_2(G,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(G,k14_polynom1(A),C) & m2_relset_1(G,k14_polynom1(A),u1_struct_0(C)) & r2_hidden(G,F) & r4_polyred(A,B,C,D,G,E) ) ) ) ) ) ) ) ) ), file(polyred,d7_polyred), []). fof(d7_termord,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) ) => ! [C] : ( ( ~ v3_struct_0(C) & l2_struct_0(C) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(D,k14_polynom1(A),C) & m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) ) => k4_termord(A,B,C,D) = k15_polynom1(A,C,D,k3_termord(A,B,C,D)) ) ) ) ) ), file(termord,d7_termord), []). fof(d8_polyred,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( ( v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) ) => ! [C] : ( ( v2_group_1(C) & v4_group_1(C) & v7_group_1(C) & v4_rlvect_1(C) & v5_rlvect_1(C) & v6_rlvect_1(C) & v7_vectsp_1(C) & v9_vectsp_1(C) & ~ v3_realset2(C) & l3_vectsp_1(C) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(D,k14_polynom1(A),C) & m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) ) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(E,k14_polynom1(A),C) & m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) ) => ( r6_polyred(A,B,C,D,E) <=> ? [F] : ( v1_funct_1(F) & v1_funct_2(F,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(F,k14_polynom1(A),C) & m2_relset_1(F,k14_polynom1(A),u1_struct_0(C)) & r4_polyred(A,B,C,D,E,F) ) ) ) ) ) ) ) ), file(polyred,d8_polyred), []). fof(d9_polynom1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( l2_struct_0(B) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,A,u1_struct_0(B)) & m2_relset_1(C,A,u1_struct_0(B)) ) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(A)) => ( D = k12_polynom1(A,B,C) <=> ! [E] : ( m1_subset_1(E,A) => ( r2_hidden(E,D) <=> k8_funct_2(A,u1_struct_0(B),C,E) != k1_rlvect_1(B) ) ) ) ) ) ) ) ), file(polynom1,d9_polynom1), []). fof(d9_polyred,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( ( v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) ) => ! [C] : ( ( v2_group_1(C) & v4_group_1(C) & v7_group_1(C) & v4_rlvect_1(C) & v5_rlvect_1(C) & v6_rlvect_1(C) & v7_vectsp_1(C) & v9_vectsp_1(C) & ~ v3_realset2(C) & l3_vectsp_1(C) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(D,k14_polynom1(A),C) & m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) ) => ! [E] : ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) => ( r7_polyred(A,B,C,D,E) <=> ? [F] : ( v1_funct_1(F) & v1_funct_2(F,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(F,k14_polynom1(A),C) & m2_relset_1(F,k14_polynom1(A),u1_struct_0(C)) & r5_polyred(A,B,C,D,F,E) ) ) ) ) ) ) ) ), file(polyred,d9_polyred), []). fof(dt_g3_vectsp_1,axiom,( ! [A,B,C,D,E] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),A) & m1_relset_1(B,k2_zfmisc_1(A,A),A) & v1_funct_1(C) & v1_funct_2(C,k2_zfmisc_1(A,A),A) & m1_relset_1(C,k2_zfmisc_1(A,A),A) & m1_subset_1(D,A) & m1_subset_1(E,A) ) => ( v3_vectsp_1(g3_vectsp_1(A,B,C,D,E)) & l3_vectsp_1(g3_vectsp_1(A,B,C,D,E)) ) ) ), file(vectsp_1,g3_vectsp_1), []). fof(dt_k12_polynom1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & l2_struct_0(B) & v1_funct_1(C) & v1_funct_2(C,A,u1_struct_0(B)) & m1_relset_1(C,A,u1_struct_0(B)) ) => m1_subset_1(k12_polynom1(A,B,C),k1_zfmisc_1(A)) ) ), file(polynom1,k12_polynom1), []). fof(dt_k13_ideal_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) ) => m1_subset_1(k13_ideal_1(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ), file(ideal_1,k13_ideal_1), []). fof(dt_k13_polynom1,axiom,( $true ), file(polynom1,k13_polynom1), []). fof(dt_k14_polynom1,axiom,( ! [A] : m1_subset_1(k14_polynom1(A),k1_zfmisc_1(k13_polynom1(A))) ), file(polynom1,k14_polynom1), []). fof(dt_k15_polynom1,axiom,( ! [A,B,C,D] : ( ( ~ v3_struct_0(B) & l1_struct_0(B) & v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) & v7_seqm_3(D) & v1_polynom1(D) & m1_pboole(D,A) ) => m1_subset_1(k15_polynom1(A,B,C,D),u1_struct_0(B)) ) ), file(polynom1,k15_polynom1), []). fof(dt_k16_polynom1,axiom,( ! [A] : m1_polynom1(k16_polynom1(A),A,k14_polynom1(A)) ), file(polynom1,k16_polynom1), []). fof(dt_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), []). fof(dt_k1_groeb_1,axiom,( ! [A,B,C] : ( ( v3_ordinal1(A) & v2_group_1(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v7_vectsp_1(B) & ~ v3_realset2(B) & l3_vectsp_1(B) & v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(C,k14_polynom1(A),B) & m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) ) => ( ~ v1_xboole_0(k1_groeb_1(A,B,C)) & v1_finset_1(k1_groeb_1(A,B,C)) & m1_subset_1(k1_groeb_1(A,B,C),k1_zfmisc_1(u1_struct_0(k30_polynom1(A,B)))) ) ) ), file(groeb_1,k1_groeb_1), []). fof(dt_k1_group_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_group_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k1_group_1(A,B,C),u1_struct_0(A)) ) ), file(group_1,k1_group_1), []). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), []). fof(dt_k1_rlvect_1,axiom,( ! [A] : ( l2_struct_0(A) => m1_subset_1(k1_rlvect_1(A),u1_struct_0(A)) ) ), file(rlvect_1,k1_rlvect_1), []). fof(dt_k1_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_k22_polynom1,axiom,( ! [A,B,C,D] : ( ( ~ v3_struct_0(B) & l1_rlvect_1(B) & v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) & v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(B)) & m1_relset_1(D,k14_polynom1(A),u1_struct_0(B)) ) => ( v1_funct_1(k22_polynom1(A,B,C,D)) & v1_funct_2(k22_polynom1(A,B,C,D),k14_polynom1(A),u1_struct_0(B)) & m2_relset_1(k22_polynom1(A,B,C,D),k14_polynom1(A),u1_struct_0(B)) ) ) ), file(polynom1,k22_polynom1), []). fof(dt_k23_polynom1,axiom,( ! [A,B,C,D] : ( ( ~ v3_struct_0(B) & v3_rlvect_1(B) & v5_rlvect_1(B) & l1_rlvect_1(B) & v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) & v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(B)) & m1_relset_1(D,k14_polynom1(A),u1_struct_0(B)) ) => ( v1_funct_1(k23_polynom1(A,B,C,D)) & v1_funct_2(k23_polynom1(A,B,C,D),k14_polynom1(A),u1_struct_0(B)) & m2_relset_1(k23_polynom1(A,B,C,D),k14_polynom1(A),u1_struct_0(B)) ) ) ), file(polynom1,k23_polynom1), []). fof(dt_k24_polynom1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & l1_rlvect_1(B) & v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) ) => ( v1_funct_1(k24_polynom1(A,B,C)) & v1_funct_2(k24_polynom1(A,B,C),k14_polynom1(A),u1_struct_0(B)) & m2_relset_1(k24_polynom1(A,B,C),k14_polynom1(A),u1_struct_0(B)) ) ) ), file(polynom1,k24_polynom1), []). fof(dt_k25_polynom1,axiom,( ! [A,B,C,D] : ( ( ~ v3_struct_0(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & l1_rlvect_1(B) & v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) & v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(B)) & m1_relset_1(D,k14_polynom1(A),u1_struct_0(B)) ) => ( v1_funct_1(k25_polynom1(A,B,C,D)) & v1_funct_2(k25_polynom1(A,B,C,D),k14_polynom1(A),u1_struct_0(B)) & m2_relset_1(k25_polynom1(A,B,C,D),k14_polynom1(A),u1_struct_0(B)) ) ) ), file(polynom1,k25_polynom1), []). fof(dt_k26_polynom1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & l2_struct_0(B) ) => ( v1_funct_1(k26_polynom1(A,B)) & v1_funct_2(k26_polynom1(A,B),k14_polynom1(A),u1_struct_0(B)) & m2_relset_1(k26_polynom1(A,B),k14_polynom1(A),u1_struct_0(B)) ) ) ), file(polynom1,k26_polynom1), []). fof(dt_k27_polynom1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & v2_group_1(B) & l2_vectsp_1(B) ) => ( v1_funct_1(k27_polynom1(A,B)) & v1_funct_2(k27_polynom1(A,B),k14_polynom1(A),u1_struct_0(B)) & m2_relset_1(k27_polynom1(A,B),k14_polynom1(A),u1_struct_0(B)) ) ) ), file(polynom1,k27_polynom1), []). fof(dt_k28_polynom1,axiom,( ! [A,B,C,D] : ( ( v3_ordinal1(A) & ~ v3_struct_0(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & l3_vectsp_1(B) & v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) & v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(B)) & m1_relset_1(D,k14_polynom1(A),u1_struct_0(B)) ) => ( v1_funct_1(k28_polynom1(A,B,C,D)) & v1_funct_2(k28_polynom1(A,B,C,D),k14_polynom1(A),u1_struct_0(B)) & m2_relset_1(k28_polynom1(A,B,C,D),k14_polynom1(A),u1_struct_0(B)) ) ) ), file(polynom1,k28_polynom1), []). fof(dt_k2_groeb_1,axiom,( ! [A,B,C,D] : ( ( v3_ordinal1(A) & v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & m1_relset_1(B,k14_polynom1(A),k14_polynom1(A)) & ~ v3_struct_0(C) & v2_group_1(C) & v4_rlvect_1(C) & v5_rlvect_1(C) & v6_rlvect_1(C) & v7_vectsp_1(C) & ~ v3_realset2(C) & l3_vectsp_1(C) & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) ) => m1_subset_1(k2_groeb_1(A,B,C,D),k1_zfmisc_1(k14_polynom1(A))) ) ), file(groeb_1,k2_groeb_1), []). fof(dt_k2_group_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_group_1(A) ) => m1_subset_1(k2_group_1(A),u1_struct_0(A)) ) ), file(group_1,k2_group_1), []). fof(dt_k2_rlvect_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_rlvect_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k2_rlvect_1(A,B,C),u1_struct_0(A)) ) ), file(rlvect_1,k2_rlvect_1), []). fof(dt_k2_termord,axiom,( ! [A,B,C,D] : ( ( v3_ordinal1(A) & v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_2(B) & v4_relat_2(B) & v8_relat_2(B) & m1_relset_1(B,k14_polynom1(A),k14_polynom1(A)) & v7_seqm_3(C) & v1_polynom1(C) & m1_pboole(C,A) & v7_seqm_3(D) & v1_polynom1(D) & m1_pboole(D,A) ) => ( v7_seqm_3(k2_termord(A,B,C,D)) & v1_polynom1(k2_termord(A,B,C,D)) & m1_pboole(k2_termord(A,B,C,D),A) ) ) ), file(termord,k2_termord), []). fof(dt_k2_vectsp_1,axiom,( ! [A] : ( l1_vectsp_1(A) => m1_subset_1(k2_vectsp_1(A),u1_struct_0(A)) ) ), file(vectsp_1,k2_vectsp_1), []). fof(dt_k2_xboole_0,axiom,( $true ), file(xboole_0,k2_xboole_0), []). fof(dt_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), []). fof(dt_k30_polynom1,axiom,( ! [A,B] : ( ( v3_ordinal1(A) & ~ v3_struct_0(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v2_group_1(B) & v7_vectsp_1(B) & ~ v3_realset2(B) & l3_vectsp_1(B) ) => ( ~ v3_struct_0(k30_polynom1(A,B)) & v3_vectsp_1(k30_polynom1(A,B)) & l3_vectsp_1(k30_polynom1(A,B)) ) ) ), file(polynom1,k30_polynom1), []). fof(dt_k3_polyred,axiom,( ! [A,B,C,D] : ( ( v3_ordinal1(A) & v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & m1_relset_1(B,k14_polynom1(A),k14_polynom1(A)) & v2_group_1(C) & v4_group_1(C) & v7_group_1(C) & v4_rlvect_1(C) & v5_rlvect_1(C) & v6_rlvect_1(C) & v7_vectsp_1(C) & v9_vectsp_1(C) & ~ v3_realset2(C) & l3_vectsp_1(C) & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) ) => m2_relset_1(k3_polyred(A,B,C,D),k4_xboole_0(u1_struct_0(k30_polynom1(A,C)),k1_tarski(k26_polynom1(A,C))),u1_struct_0(k30_polynom1(A,C))) ) ), file(polyred,k3_polyred), []). fof(dt_k3_termord,axiom,( ! [A,B,C,D] : ( ( v3_ordinal1(A) & v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & m1_relset_1(B,k14_polynom1(A),k14_polynom1(A)) & ~ v3_struct_0(C) & l2_struct_0(C) & v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(D,k14_polynom1(A),C) & m1_relset_1(D,k14_polynom1(A),u1_struct_0(C)) ) => m1_polynom1(k3_termord(A,B,C,D),A,k14_polynom1(A)) ) ), file(termord,k3_termord), []). fof(dt_k3_xboole_0,axiom,( $true ), file(xboole_0,k3_xboole_0), []). fof(dt_k4_rlvect_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & l1_rlvect_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k4_rlvect_1(A,B,C),u1_struct_0(A)) ) ), file(rlvect_1,k4_rlvect_1), []). fof(dt_k4_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_termord,axiom,( ! [A,B,C,D] : ( ( v3_ordinal1(A) & v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & m1_relset_1(B,k14_polynom1(A),k14_polynom1(A)) & ~ v3_struct_0(C) & l2_struct_0(C) & v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(D,k14_polynom1(A),C) & m1_relset_1(D,k14_polynom1(A),u1_struct_0(C)) ) => m1_subset_1(k4_termord(A,B,C,D),u1_struct_0(C)) ) ), file(termord,k4_termord), []). fof(dt_k4_xboole_0,axiom,( $true ), file(xboole_0,k4_xboole_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_rlvect_1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_rlvect_1(A) & m1_subset_1(B,u1_struct_0(A)) ) => m1_subset_1(k5_rlvect_1(A,B),u1_struct_0(A)) ) ), file(rlvect_1,k5_rlvect_1), []). fof(dt_k6_rlvect_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_rlvect_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k6_rlvect_1(A,B,C),u1_struct_0(A)) ) ), file(rlvect_1,k6_rlvect_1), []). fof(dt_k6_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_termord,axiom,( ! [A,B,C,D] : ( ( v3_ordinal1(A) & v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & m1_relset_1(B,k14_polynom1(A),k14_polynom1(A)) & ~ v3_struct_0(C) & v4_rlvect_1(C) & v5_rlvect_1(C) & v6_rlvect_1(C) & l1_rlvect_1(C) & v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(D,k14_polynom1(A),C) & m1_relset_1(D,k14_polynom1(A),u1_struct_0(C)) ) => ( v1_funct_1(k6_termord(A,B,C,D)) & v1_funct_2(k6_termord(A,B,C,D),k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(k6_termord(A,B,C,D),k14_polynom1(A),C) & m2_relset_1(k6_termord(A,B,C,D),k14_polynom1(A),u1_struct_0(C)) ) ) ), file(termord,k6_termord), []). fof(dt_k7_ideal_1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l3_vectsp_1(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( ~ v1_xboole_0(k7_ideal_1(A,B)) & v1_ideal_1(k7_ideal_1(A,B),A) & v2_ideal_1(k7_ideal_1(A,B),A) & v3_ideal_1(k7_ideal_1(A,B),A) & m1_subset_1(k7_ideal_1(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ) ), file(ideal_1,k7_ideal_1), []). fof(dt_k8_funct_2,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & m1_subset_1(D,A) ) => m1_subset_1(k8_funct_2(A,B,C,D),B) ) ), file(funct_2,k8_funct_2), []). fof(dt_l1_group_1,axiom,( ! [A] : ( l1_group_1(A) => l1_struct_0(A) ) ), file(group_1,l1_group_1), []). fof(dt_l1_rlvect_1,axiom,( ! [A] : ( l1_rlvect_1(A) => l2_struct_0(A) ) ), file(rlvect_1,l1_rlvect_1), []). fof(dt_l1_struct_0,axiom,( $true ), file(struct_0,l1_struct_0), []). fof(dt_l1_vectsp_1,axiom,( ! [A] : ( l1_vectsp_1(A) => l1_group_1(A) ) ), file(vectsp_1,l1_vectsp_1), []). fof(dt_l2_struct_0,axiom,( ! [A] : ( l2_struct_0(A) => l1_struct_0(A) ) ), file(struct_0,l2_struct_0), []). fof(dt_l2_vectsp_1,axiom,( ! [A] : ( l2_vectsp_1(A) => ( l1_vectsp_1(A) & l2_struct_0(A) ) ) ), file(vectsp_1,l2_vectsp_1), []). fof(dt_l3_vectsp_1,axiom,( ! [A] : ( l3_vectsp_1(A) => ( l1_rlvect_1(A) & l2_vectsp_1(A) ) ) ), file(vectsp_1,l3_vectsp_1), []). fof(dt_m1_pboole,axiom,( ! [A,B] : ( m1_pboole(B,A) => ( v1_relat_1(B) & v1_funct_1(B) ) ) ), file(pboole,m1_pboole), []). fof(dt_m1_polynom1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(k14_polynom1(A))) ) => ! [C] : ( m1_polynom1(C,A,B) => ( v7_seqm_3(C) & v1_polynom1(C) & m1_pboole(C,A) ) ) ) ), file(polynom1,m1_polynom1), []). 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_group_1,axiom,( ! [A] : ( l1_group_1(A) => ( v1_funct_1(u1_group_1(A)) & v1_funct_2(u1_group_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) & m2_relset_1(u1_group_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) ) ) ), file(group_1,u1_group_1), []). fof(dt_u1_rlvect_1,axiom,( ! [A] : ( l1_rlvect_1(A) => ( v1_funct_1(u1_rlvect_1(A)) & v1_funct_2(u1_rlvect_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) & m2_relset_1(u1_rlvect_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) ) ) ), file(rlvect_1,u1_rlvect_1), []). fof(dt_u1_struct_0,axiom,( $true ), file(struct_0,u1_struct_0), []). fof(dt_u1_vectsp_1,axiom,( ! [A] : ( l1_vectsp_1(A) => m1_subset_1(u1_vectsp_1(A),u1_struct_0(A)) ) ), file(vectsp_1,u1_vectsp_1), []). fof(dt_u2_struct_0,axiom,( ! [A] : ( l2_struct_0(A) => m1_subset_1(u2_struct_0(A),u1_struct_0(A)) ) ), file(struct_0,u2_struct_0), []). fof(existence_l1_group_1,axiom,( ? [A] : l1_group_1(A) ), file(group_1,l1_group_1), []). fof(existence_l1_rlvect_1,axiom,( ? [A] : l1_rlvect_1(A) ), file(rlvect_1,l1_rlvect_1), []). fof(existence_l1_struct_0,axiom,( ? [A] : l1_struct_0(A) ), file(struct_0,l1_struct_0), []). fof(existence_l1_vectsp_1,axiom,( ? [A] : l1_vectsp_1(A) ), file(vectsp_1,l1_vectsp_1), []). fof(existence_l2_struct_0,axiom,( ? [A] : l2_struct_0(A) ), file(struct_0,l2_struct_0), []). fof(existence_l2_vectsp_1,axiom,( ? [A] : l2_vectsp_1(A) ), file(vectsp_1,l2_vectsp_1), []). fof(existence_l3_vectsp_1,axiom,( ? [A] : l3_vectsp_1(A) ), file(vectsp_1,l3_vectsp_1), []). fof(existence_m1_pboole,axiom,( ! [A] : ? [B] : m1_pboole(B,A) ), file(pboole,m1_pboole), []). fof(existence_m1_polynom1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(k14_polynom1(A))) ) => ? [C] : m1_polynom1(C,A,B) ) ), file(polynom1,m1_polynom1), []). 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_finset_1,axiom,( ! [A,B] : ( v1_finset_1(B) => v1_finset_1(k3_xboole_0(A,B)) ) ), file(finset_1,fc10_finset_1), []). fof(fc11_finset_1,axiom,( ! [A,B] : ( v1_finset_1(A) => v1_finset_1(k3_xboole_0(A,B)) ) ), file(finset_1,fc11_finset_1), []). 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_relat_1,axiom, ( v1_xboole_0(k1_xboole_0) & v1_relat_1(k1_xboole_0) & v3_relat_1(k1_xboole_0) ), file(relat_1,fc12_relat_1), []). 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_polynom1,axiom, ( 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_xboole_0(k1_xboole_0) & v1_finset_1(k1_xboole_0) & v1_finseq_1(k1_xboole_0) & v1_funcop_1(k1_xboole_0) & v1_membered(k1_xboole_0) & v2_membered(k1_xboole_0) & v3_membered(k1_xboole_0) & v4_membered(k1_xboole_0) & v5_membered(k1_xboole_0) & v1_seq_1(k1_xboole_0) & v1_matrlin(k1_xboole_0) & v7_seqm_3(k1_xboole_0) ), file(polynom1,fc14_polynom1), []). fof(fc15_ideal_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v5_rlvect_1(A) & v5_vectsp_1(A) & v2_algstr_1(A) & l3_vectsp_1(A) & ~ v1_xboole_0(B) & v2_ideal_1(B,A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(C) & v2_ideal_1(C,A) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) ) => ~ v1_xboole_0(k3_xboole_0(B,C)) ) ), file(ideal_1,fc15_ideal_1), []). fof(fc16_ideal_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_rlvect_1(A) & v1_ideal_1(B,A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v1_ideal_1(C,A) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) ) => v1_ideal_1(k3_xboole_0(B,C),A) ) ), file(ideal_1,fc16_ideal_1), []). fof(fc17_ideal_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_vectsp_1(A) & v2_ideal_1(B,A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v2_ideal_1(C,A) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) ) => v2_ideal_1(k3_xboole_0(B,C),A) ) ), file(ideal_1,fc17_ideal_1), []). fof(fc18_ideal_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_vectsp_1(A) & v3_ideal_1(B,A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v3_ideal_1(C,A) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) ) => v3_ideal_1(k3_xboole_0(B,C),A) ) ), file(ideal_1,fc18_ideal_1), []). 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_ideal_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v5_rlvect_1(A) & l1_rlvect_1(A) ) => v1_ideal_1(k1_tarski(k1_rlvect_1(A)),A) ) ), file(ideal_1,fc1_ideal_1), []). 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_polynom1,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(A) & v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),A) & m1_relset_1(B,k2_zfmisc_1(A,A),A) & v1_funct_1(C) & v1_funct_2(C,k2_zfmisc_1(A,A),A) & m1_relset_1(C,k2_zfmisc_1(A,A),A) & m1_subset_1(D,A) & m1_subset_1(E,A) ) => ( ~ v3_struct_0(g3_vectsp_1(A,B,C,D,E)) & v3_vectsp_1(g3_vectsp_1(A,B,C,D,E)) ) ) ), file(polynom1,fc1_polynom1), []). fof(fc1_polynom2,axiom,( ! [A,B,C,D] : ( ( v3_ordinal1(A) & ~ v3_struct_0(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & l1_rlvect_1(B) & v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(C,k14_polynom1(A),B) & m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) & v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(D,k14_polynom1(A),B) & m1_relset_1(D,k14_polynom1(A),u1_struct_0(B)) ) => ( v1_relat_1(k25_polynom1(A,B,C,D)) & v1_funct_1(k25_polynom1(A,B,C,D)) & v1_funct_2(k25_polynom1(A,B,C,D),k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(k25_polynom1(A,B,C,D),k14_polynom1(A),B) ) ) ), file(polynom2,fc1_polynom2), []). fof(fc1_polynom7,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & l2_struct_0(B) ) => ( v1_relat_1(k26_polynom1(A,B)) & v1_funct_1(k26_polynom1(A,B)) & v1_funct_2(k26_polynom1(A,B),k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(k26_polynom1(A,B),k14_polynom1(A),B) & v3_polynom7(k26_polynom1(A,B),A,B) & v4_polynom7(k26_polynom1(A,B),A,B) ) ) ), file(polynom7,fc1_polynom7), []). fof(fc1_polyred,axiom,( ! [A,B,C,D] : ( ( v3_ordinal1(A) & v2_group_1(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v7_vectsp_1(B) & v2_vectsp_2(B) & v1_algstr_1(B) & ~ v3_realset2(B) & l3_vectsp_1(B) & v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(C,k14_polynom1(A),B) & v1_polynom7(C,A,B) & m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) & v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(D,k14_polynom1(A),B) & v1_polynom7(D,A,B) & m1_relset_1(D,k14_polynom1(A),u1_struct_0(B)) ) => ( v1_relat_1(k28_polynom1(A,B,C,D)) & v1_funct_1(k28_polynom1(A,B,C,D)) & v1_funct_2(k28_polynom1(A,B,C,D),k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(k28_polynom1(A,B,C,D),k14_polynom1(A),B) & v1_polynom7(k28_polynom1(A,B,C,D),A,B) ) ) ), file(polyred,fc1_polyred), []). fof(fc1_relat_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_relat_1(B) ) => v1_relat_1(k3_xboole_0(A,B)) ) ), file(relat_1,fc1_relat_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_termord,axiom,( ! [A,B,C,D] : ( ( v3_ordinal1(A) & ~ v3_struct_0(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v2_group_1(B) & v7_vectsp_1(B) & l3_vectsp_1(B) & v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & v3_polynom7(C,A,B) & m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) & v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(B)) & v3_polynom7(D,A,B) & m1_relset_1(D,k14_polynom1(A),u1_struct_0(B)) ) => ( v1_relat_1(k28_polynom1(A,B,C,D)) & v1_funct_1(k28_polynom1(A,B,C,D)) & v1_funct_2(k28_polynom1(A,B,C,D),k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(k28_polynom1(A,B,C,D),k14_polynom1(A),B) & v3_polynom7(k28_polynom1(A,B,C,D),A,B) ) ) ), file(termord,fc1_termord), []). fof(fc20_polynom1,axiom, ( 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_xboole_0(k1_xboole_0) & v1_finset_1(k1_xboole_0) & v1_finseq_1(k1_xboole_0) & v1_funcop_1(k1_xboole_0) & v1_membered(k1_xboole_0) & v2_membered(k1_xboole_0) & v3_membered(k1_xboole_0) & v4_membered(k1_xboole_0) & v5_membered(k1_xboole_0) & v1_seq_1(k1_xboole_0) & v1_matrlin(k1_xboole_0) & v7_seqm_3(k1_xboole_0) & v1_polynom1(k1_xboole_0) ), file(polynom1,fc20_polynom1), []). fof(fc25_polynom1,axiom,( ! [A] : ~ v1_xboole_0(k13_polynom1(A)) ), file(polynom1,fc25_polynom1), []). fof(fc2_ideal_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v4_vectsp_1(A) & v1_algstr_1(A) & v3_algstr_1(A) & l3_vectsp_1(A) ) => v2_ideal_1(k1_tarski(k1_rlvect_1(A)),A) ) ), file(ideal_1,fc2_ideal_1), []). fof(fc2_ordinal1,axiom, ( 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_xboole_0(k1_xboole_0) & v1_ordinal1(k1_xboole_0) & v2_ordinal1(k1_xboole_0) & v3_ordinal1(k1_xboole_0) ), file(ordinal1,fc2_ordinal1), []). fof(fc2_polynom2,axiom,( ! [A,B,C] : ( ( v3_ordinal1(A) & ~ v3_struct_0(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v2_group_1(B) & v7_vectsp_1(B) & ~ v3_realset2(B) & l3_vectsp_1(B) & v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(C,k14_polynom1(A),B) & m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) ) => v1_finset_1(k12_polynom1(k14_polynom1(A),B,C)) ) ), file(polynom2,fc2_polynom2), []). fof(fc2_polynom7,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & v2_group_1(B) & l3_vectsp_1(B) ) => ( v1_relat_1(k27_polynom1(A,B)) & v1_funct_1(k27_polynom1(A,B)) & v1_funct_2(k27_polynom1(A,B),k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(k27_polynom1(A,B),k14_polynom1(A),B) & v3_polynom7(k27_polynom1(A,B),A,B) & v4_polynom7(k27_polynom1(A,B),A,B) ) ) ), file(polynom7,fc2_polynom7), []). fof(fc2_relat_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_relat_1(B) ) => v1_relat_1(k2_xboole_0(A,B)) ) ), file(relat_1,fc2_relat_1), []). fof(fc2_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_tarski(A)) ), file(subset_1,fc2_subset_1), []). fof(fc2_termord,axiom,( ! [A,B,C,D] : ( ( v3_ordinal1(A) & ~ v3_struct_0(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v7_vectsp_1(B) & l3_vectsp_1(B) & v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & v4_polynom7(C,A,B) & m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) & v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(B)) & v4_polynom7(D,A,B) & m1_relset_1(D,k14_polynom1(A),u1_struct_0(B)) ) => ( v1_relat_1(k28_polynom1(A,B,C,D)) & v1_funct_1(k28_polynom1(A,B,C,D)) & v1_funct_2(k28_polynom1(A,B,C,D),k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(k28_polynom1(A,B,C,D),k14_polynom1(A),B) & v3_polynom7(k28_polynom1(A,B,C,D),A,B) & v4_polynom7(k28_polynom1(A,B,C,D),A,B) ) ) ), file(termord,fc2_termord), []). fof(fc31_ideal_1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l3_vectsp_1(A) & ~ v1_xboole_0(B) & v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( ~ v1_xboole_0(k7_ideal_1(A,B)) & v1_ideal_1(k7_ideal_1(A,B),A) & v2_ideal_1(k7_ideal_1(A,B),A) & v3_ideal_1(k7_ideal_1(A,B),A) & v6_ideal_1(k7_ideal_1(A,B),A) ) ) ), file(ideal_1,fc31_ideal_1), []). fof(fc32_polynom1,axiom,( ! [A,B,C,D] : ( ( v3_ordinal1(A) & ~ v3_struct_0(B) & v5_rlvect_1(B) & l1_rlvect_1(B) & v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(C,k14_polynom1(A),B) & m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) & v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(D,k14_polynom1(A),B) & m1_relset_1(D,k14_polynom1(A),u1_struct_0(B)) ) => ( v1_relat_1(k22_polynom1(A,B,C,D)) & v1_funct_1(k22_polynom1(A,B,C,D)) & v1_funct_2(k22_polynom1(A,B,C,D),k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(k22_polynom1(A,B,C,D),k14_polynom1(A),B) ) ) ), file(polynom1,fc32_polynom1), []). fof(fc33_polynom1,axiom,( ! [A,B,C] : ( ( v3_ordinal1(A) & ~ v3_struct_0(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & l1_rlvect_1(B) & v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(C,k14_polynom1(A),B) & m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) ) => ( v1_relat_1(k24_polynom1(A,B,C)) & v1_funct_1(k24_polynom1(A,B,C)) & v1_funct_2(k24_polynom1(A,B,C),k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(k24_polynom1(A,B,C),k14_polynom1(A),B) ) ) ), file(polynom1,fc33_polynom1), []). fof(fc34_polynom1,axiom,( ! [A,B,C,D] : ( ( m1_subset_1(A,k5_numbers) & ~ v3_struct_0(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & l1_rlvect_1(B) & v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(C,k14_polynom1(A),B) & m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) & v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(D,k14_polynom1(A),B) & m1_relset_1(D,k14_polynom1(A),u1_struct_0(B)) ) => ( v1_relat_1(k25_polynom1(A,B,C,D)) & v1_funct_1(k25_polynom1(A,B,C,D)) & v1_funct_2(k25_polynom1(A,B,C,D),k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(k25_polynom1(A,B,C,D),k14_polynom1(A),B) ) ) ), file(polynom1,fc34_polynom1), []). fof(fc35_polynom1,axiom,( ! [A,B] : ( ( v3_ordinal1(A) & ~ v3_struct_0(B) & l2_struct_0(B) ) => ( v1_relat_1(k26_polynom1(A,B)) & v1_funct_1(k26_polynom1(A,B)) & v1_funct_2(k26_polynom1(A,B),k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(k26_polynom1(A,B),k14_polynom1(A),B) ) ) ), file(polynom1,fc35_polynom1), []). fof(fc36_polynom1,axiom,( ! [A,B] : ( ( v3_ordinal1(A) & ~ v3_struct_0(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v2_group_1(B) & v4_vectsp_1(B) & ~ v3_realset2(B) & l3_vectsp_1(B) ) => ( v1_relat_1(k27_polynom1(A,B)) & v1_funct_1(k27_polynom1(A,B)) & v1_funct_2(k27_polynom1(A,B),k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(k27_polynom1(A,B),k14_polynom1(A),B) ) ) ), file(polynom1,fc36_polynom1), []). fof(fc37_polynom1,axiom,( ! [A,B,C,D] : ( ( v3_ordinal1(A) & ~ v3_struct_0(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v2_group_1(B) & v7_vectsp_1(B) & l3_vectsp_1(B) & v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(C,k14_polynom1(A),B) & m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) & v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(D,k14_polynom1(A),B) & m1_relset_1(D,k14_polynom1(A),u1_struct_0(B)) ) => ( v1_relat_1(k28_polynom1(A,B,C,D)) & v1_funct_1(k28_polynom1(A,B,C,D)) & v1_funct_2(k28_polynom1(A,B,C,D),k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(k28_polynom1(A,B,C,D),k14_polynom1(A),B) ) ) ), file(polynom1,fc37_polynom1), []). fof(fc38_polynom1,axiom,( ! [A,B] : ( ( v3_ordinal1(A) & ~ v3_struct_0(B) & v3_rlvect_1(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v2_group_1(B) & v7_vectsp_1(B) & ~ v3_realset2(B) & l3_vectsp_1(B) ) => ( ~ v3_struct_0(k30_polynom1(A,B)) & v3_rlvect_1(k30_polynom1(A,B)) & v3_vectsp_1(k30_polynom1(A,B)) ) ) ), file(polynom1,fc38_polynom1), []). fof(fc39_polynom1,axiom,( ! [A,B] : ( ( v3_ordinal1(A) & ~ v3_struct_0(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v2_group_1(B) & v7_vectsp_1(B) & ~ v3_realset2(B) & l3_vectsp_1(B) ) => ( ~ v3_struct_0(k30_polynom1(A,B)) & v4_rlvect_1(k30_polynom1(A,B)) & v3_vectsp_1(k30_polynom1(A,B)) ) ) ), file(polynom1,fc39_polynom1), []). fof(fc3_ideal_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v5_rlvect_1(A) & v5_vectsp_1(A) & v2_algstr_1(A) & l3_vectsp_1(A) ) => v3_ideal_1(k1_tarski(k1_rlvect_1(A)),A) ) ), file(ideal_1,fc3_ideal_1), []). fof(fc3_polynom1,axiom, ( 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_xboole_0(k1_xboole_0) & v1_finset_1(k1_xboole_0) & v1_finseq_1(k1_xboole_0) & v1_funcop_1(k1_xboole_0) & v1_membered(k1_xboole_0) & v2_membered(k1_xboole_0) & v3_membered(k1_xboole_0) & v4_membered(k1_xboole_0) & v5_membered(k1_xboole_0) & v1_matrlin(k1_xboole_0) ), file(polynom1,fc3_polynom1), []). fof(fc3_polynom2,axiom,( ! [A,B] : ( ( v3_ordinal1(A) & ~ v3_struct_0(B) & v3_rlvect_1(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v2_group_1(B) & v4_group_1(B) & v7_vectsp_1(B) & ~ v3_realset2(B) & l3_vectsp_1(B) ) => ( ~ v3_struct_0(k30_polynom1(A,B)) & v3_rlvect_1(k30_polynom1(A,B)) & v4_rlvect_1(k30_polynom1(A,B)) & v5_rlvect_1(k30_polynom1(A,B)) & v6_rlvect_1(k30_polynom1(A,B)) & v2_group_1(k30_polynom1(A,B)) & v4_group_1(k30_polynom1(A,B)) & v3_vectsp_1(k30_polynom1(A,B)) & v4_vectsp_1(k30_polynom1(A,B)) & v6_vectsp_1(k30_polynom1(A,B)) & v8_vectsp_1(k30_polynom1(A,B)) & v1_algstr_1(k30_polynom1(A,B)) & v2_algstr_1(k30_polynom1(A,B)) & v3_algstr_1(k30_polynom1(A,B)) & v4_algstr_1(k30_polynom1(A,B)) & v5_algstr_1(k30_polynom1(A,B)) & v6_algstr_1(k30_polynom1(A,B)) ) ) ), file(polynom2,fc3_polynom2), []). fof(fc3_relat_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_relat_1(B) ) => v1_relat_1(k4_xboole_0(A,B)) ) ), file(relat_1,fc3_relat_1), []). fof(fc40_polynom1,axiom,( ! [A,B] : ( ( v3_ordinal1(A) & ~ v3_struct_0(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v2_group_1(B) & v7_vectsp_1(B) & ~ v3_realset2(B) & l3_vectsp_1(B) ) => ( ~ v3_struct_0(k30_polynom1(A,B)) & v4_rlvect_1(k30_polynom1(A,B)) & v5_rlvect_1(k30_polynom1(A,B)) & v3_vectsp_1(k30_polynom1(A,B)) ) ) ), file(polynom1,fc40_polynom1), []). fof(fc41_polynom1,axiom,( ! [A,B] : ( ( v3_ordinal1(A) & ~ v3_struct_0(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v2_group_1(B) & v7_vectsp_1(B) & ~ v3_realset2(B) & l3_vectsp_1(B) ) => ( ~ v3_struct_0(k30_polynom1(A,B)) & v4_rlvect_1(k30_polynom1(A,B)) & v5_rlvect_1(k30_polynom1(A,B)) & v6_rlvect_1(k30_polynom1(A,B)) & v3_vectsp_1(k30_polynom1(A,B)) & v1_algstr_1(k30_polynom1(A,B)) & v2_algstr_1(k30_polynom1(A,B)) & v3_algstr_1(k30_polynom1(A,B)) & v4_algstr_1(k30_polynom1(A,B)) & v5_algstr_1(k30_polynom1(A,B)) & v6_algstr_1(k30_polynom1(A,B)) ) ) ), file(polynom1,fc41_polynom1), []). fof(fc42_polynom1,axiom,( ! [A,B] : ( ( v3_ordinal1(A) & ~ v3_struct_0(B) & v3_rlvect_1(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v2_group_1(B) & v7_group_1(B) & v7_vectsp_1(B) & ~ v3_realset2(B) & l3_vectsp_1(B) ) => ( ~ v3_struct_0(k30_polynom1(A,B)) & v3_rlvect_1(k30_polynom1(A,B)) & v4_rlvect_1(k30_polynom1(A,B)) & v5_rlvect_1(k30_polynom1(A,B)) & v6_rlvect_1(k30_polynom1(A,B)) & v7_group_1(k30_polynom1(A,B)) & v3_vectsp_1(k30_polynom1(A,B)) & v1_algstr_1(k30_polynom1(A,B)) & v2_algstr_1(k30_polynom1(A,B)) & v3_algstr_1(k30_polynom1(A,B)) & v4_algstr_1(k30_polynom1(A,B)) & v5_algstr_1(k30_polynom1(A,B)) & v6_algstr_1(k30_polynom1(A,B)) ) ) ), file(polynom1,fc42_polynom1), []). fof(fc43_polynom1,axiom,( ! [A,B] : ( ( v3_ordinal1(A) & ~ v3_struct_0(B) & v3_rlvect_1(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v2_group_1(B) & v4_group_1(B) & v7_vectsp_1(B) & ~ v3_realset2(B) & l3_vectsp_1(B) ) => ( ~ v3_struct_0(k30_polynom1(A,B)) & v3_rlvect_1(k30_polynom1(A,B)) & v4_rlvect_1(k30_polynom1(A,B)) & v5_rlvect_1(k30_polynom1(A,B)) & v6_rlvect_1(k30_polynom1(A,B)) & v4_group_1(k30_polynom1(A,B)) & v3_vectsp_1(k30_polynom1(A,B)) & v1_algstr_1(k30_polynom1(A,B)) & v2_algstr_1(k30_polynom1(A,B)) & v3_algstr_1(k30_polynom1(A,B)) & v4_algstr_1(k30_polynom1(A,B)) & v5_algstr_1(k30_polynom1(A,B)) & v6_algstr_1(k30_polynom1(A,B)) ) ) ), file(polynom1,fc43_polynom1), []). fof(fc44_polynom1,axiom,( ! [A,B] : ( ( v3_ordinal1(A) & ~ v3_struct_0(B) & v3_rlvect_1(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v2_group_1(B) & v4_group_1(B) & v7_vectsp_1(B) & ~ v3_realset2(B) & l3_vectsp_1(B) ) => ( ~ v3_struct_0(k30_polynom1(A,B)) & v3_rlvect_1(k30_polynom1(A,B)) & v4_rlvect_1(k30_polynom1(A,B)) & v5_rlvect_1(k30_polynom1(A,B)) & v6_rlvect_1(k30_polynom1(A,B)) & v2_group_1(k30_polynom1(A,B)) & v4_group_1(k30_polynom1(A,B)) & v3_vectsp_1(k30_polynom1(A,B)) & v4_vectsp_1(k30_polynom1(A,B)) & v6_vectsp_1(k30_polynom1(A,B)) & v8_vectsp_1(k30_polynom1(A,B)) & v1_algstr_1(k30_polynom1(A,B)) & v2_algstr_1(k30_polynom1(A,B)) & v3_algstr_1(k30_polynom1(A,B)) & v4_algstr_1(k30_polynom1(A,B)) & v5_algstr_1(k30_polynom1(A,B)) & v6_algstr_1(k30_polynom1(A,B)) ) ) ), file(polynom1,fc44_polynom1), []). fof(fc4_relat_1,axiom, ( v1_xboole_0(k1_xboole_0) & v1_relat_1(k1_xboole_0) ), file(relat_1,fc4_relat_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(fc4_termord,axiom,( ! [A,B,C,D] : ( ( v3_ordinal1(A) & v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & m1_relset_1(B,k14_polynom1(A),k14_polynom1(A)) & ~ v3_realset2(C) & l2_struct_0(C) & v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(D,k14_polynom1(A),C) & v1_polynom7(D,A,C) & m1_relset_1(D,k14_polynom1(A),u1_struct_0(C)) ) => v8_rlvect_1(k4_termord(A,B,C,D),C) ) ), file(termord,fc4_termord), []). fof(fc5_polynom1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & v1_matrlin(A) ) => ( v1_relat_1(k1_funct_1(A,B)) & v1_funct_1(k1_funct_1(A,B)) & v1_finset_1(k1_funct_1(A,B)) & v1_finseq_1(k1_funct_1(A,B)) ) ) ), file(polynom1,fc5_polynom1), []). fof(fc5_polyred,axiom,( ! [A,B,C,D] : ( ( m1_subset_1(A,k5_numbers) & v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v2_bagorder(B,A) & m1_relset_1(B,k14_polynom1(A),k14_polynom1(A)) & ~ v3_struct_0(C) & v2_group_1(C) & v4_group_1(C) & v7_group_1(C) & v3_rlvect_1(C) & v4_rlvect_1(C) & v5_rlvect_1(C) & v6_rlvect_1(C) & v7_vectsp_1(C) & v9_vectsp_1(C) & ~ v10_vectsp_1(C) & l3_vectsp_1(C) & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) ) => ( v1_relat_1(k3_polyred(A,B,C,D)) & v2_relat_2(k3_polyred(A,B,C,D)) & v1_rewrite1(k3_polyred(A,B,C,D)) & v2_rewrite1(k3_polyred(A,B,C,D)) & v3_rewrite1(k3_polyred(A,B,C,D)) ) ) ), file(polyred,fc5_polyred), []). 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(fraenkel_a_4_0_groeb_1,axiom,( ! [A,B,C,D,E] : ( ( v3_ordinal1(B) & v1_partfun1(C,k14_polynom1(B),k14_polynom1(B)) & v1_relat_2(C) & v4_relat_2(C) & v6_relat_2(C) & v8_relat_2(C) & m2_relset_1(C,k14_polynom1(B),k14_polynom1(B)) & ~ v3_struct_0(D) & v2_group_1(D) & v4_rlvect_1(D) & v5_rlvect_1(D) & v6_rlvect_1(D) & v7_vectsp_1(D) & ~ v3_realset2(D) & l3_vectsp_1(D) & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(B,D)))) ) => ( r2_hidden(A,a_4_0_groeb_1(B,C,D,E)) <=> ? [F] : ( v1_funct_1(F) & v1_funct_2(F,k14_polynom1(B),u1_struct_0(D)) & v2_polynom1(F,k14_polynom1(B),D) & m2_relset_1(F,k14_polynom1(B),u1_struct_0(D)) & A = k3_termord(B,C,D,F) & r2_hidden(F,E) & F != k26_polynom1(B,D) ) ) ) ), file(groeb_1,a_4_0_groeb_1), []). fof(fraenkel_a_4_11_groeb_1,axiom,( ! [A,B,C,D,E] : ( ( m2_subset_1(B,k1_numbers,k5_numbers) & v1_partfun1(C,k14_polynom1(B),k14_polynom1(B)) & v1_relat_2(C) & v4_relat_2(C) & v6_relat_2(C) & v8_relat_2(C) & v2_bagorder(C,B) & m2_relset_1(C,k14_polynom1(B),k14_polynom1(B)) & ~ v3_struct_0(D) & v2_group_1(D) & v4_group_1(D) & v7_group_1(D) & v3_rlvect_1(D) & v4_rlvect_1(D) & v5_rlvect_1(D) & v6_rlvect_1(D) & v7_vectsp_1(D) & v9_vectsp_1(D) & ~ v10_vectsp_1(D) & l3_vectsp_1(D) & ~ v1_xboole_0(E) & v1_finset_1(E) & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(B,D)))) ) => ( r2_hidden(A,a_4_11_groeb_1(B,C,D,E)) <=> ? [F] : ( v1_funct_1(F) & v1_funct_2(F,k14_polynom1(B),u1_struct_0(D)) & v2_polynom1(F,k14_polynom1(B),D) & m2_relset_1(F,k14_polynom1(B),u1_struct_0(D)) & A = k3_termord(B,C,D,F) & r2_hidden(F,E) & F != k26_polynom1(B,D) ) ) ) ), file(groeb_1,a_4_11_groeb_1), []). fof(fraenkel_a_4_12_groeb_1,axiom,( ! [A,B,C,D,E] : ( ( m2_subset_1(B,k1_numbers,k5_numbers) & v1_partfun1(C,k14_polynom1(B),k14_polynom1(B)) & v1_relat_2(C) & v4_relat_2(C) & v6_relat_2(C) & v8_relat_2(C) & v2_bagorder(C,B) & m2_relset_1(C,k14_polynom1(B),k14_polynom1(B)) & ~ v3_struct_0(D) & v2_group_1(D) & v4_group_1(D) & v7_group_1(D) & v3_rlvect_1(D) & v4_rlvect_1(D) & v5_rlvect_1(D) & v6_rlvect_1(D) & v7_vectsp_1(D) & v9_vectsp_1(D) & ~ v10_vectsp_1(D) & l3_vectsp_1(D) & ~ v1_xboole_0(E) & v1_finset_1(E) & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(B,D)))) ) => ( r2_hidden(A,a_4_12_groeb_1(B,C,D,E)) <=> ? [F] : ( v1_funct_1(F) & v1_funct_2(F,k14_polynom1(B),u1_struct_0(D)) & v2_polynom1(F,k14_polynom1(B),D) & m2_relset_1(F,k14_polynom1(B),u1_struct_0(D)) & A = k3_termord(B,C,D,F) & r2_hidden(F,k7_ideal_1(k30_polynom1(B,D),E)) & F != k26_polynom1(B,D) ) ) ) ), file(groeb_1,a_4_12_groeb_1), []). fof(fraenkel_a_4_8_groeb_1,axiom,( ! [A,B,C,D,E] : ( ( m2_subset_1(B,k1_numbers,k5_numbers) & v1_partfun1(C,k14_polynom1(B),k14_polynom1(B)) & v1_relat_2(C) & v4_relat_2(C) & v6_relat_2(C) & v8_relat_2(C) & v2_bagorder(C,B) & m2_relset_1(C,k14_polynom1(B),k14_polynom1(B)) & ~ v3_struct_0(D) & v2_group_1(D) & v4_group_1(D) & v7_group_1(D) & v3_rlvect_1(D) & v4_rlvect_1(D) & v5_rlvect_1(D) & v6_rlvect_1(D) & v7_vectsp_1(D) & v9_vectsp_1(D) & ~ v10_vectsp_1(D) & l3_vectsp_1(D) & ~ v1_xboole_0(E) & v1_ideal_1(E,k30_polynom1(B,D)) & v2_ideal_1(E,k30_polynom1(B,D)) & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(B,D)))) ) => ( r2_hidden(A,a_4_8_groeb_1(B,C,D,E)) <=> ? [F] : ( v1_funct_1(F) & v1_funct_2(F,k14_polynom1(B),u1_struct_0(D)) & v2_polynom1(F,k14_polynom1(B),D) & m2_relset_1(F,k14_polynom1(B),u1_struct_0(D)) & A = k3_termord(B,C,D,F) & r2_hidden(F,E) & F != k26_polynom1(B,D) ) ) ) ), file(groeb_1,a_4_8_groeb_1), []). fof(free_g3_vectsp_1,axiom,( ! [A,B,C,D,E] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),A) & m1_relset_1(B,k2_zfmisc_1(A,A),A) & v1_funct_1(C) & v1_funct_2(C,k2_zfmisc_1(A,A),A) & m1_relset_1(C,k2_zfmisc_1(A,A),A) & m1_subset_1(D,A) & m1_subset_1(E,A) ) => ! [F,G,H,I,J] : ( g3_vectsp_1(A,B,C,D,E) = g3_vectsp_1(F,G,H,I,J) => ( A = F & B = G & C = H & D = I & E = J ) ) ) ), file(vectsp_1,g3_vectsp_1), []). fof(idempotence_k13_ideal_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) ) => k13_ideal_1(A,B,B) = B ) ), file(ideal_1,k13_ideal_1), []). fof(idempotence_k2_xboole_0,axiom,( ! [A,B] : k2_xboole_0(A,A) = A ), file(xboole_0,k2_xboole_0), []). fof(idempotence_k3_xboole_0,axiom,( ! [A,B] : k3_xboole_0(A,A) = A ), file(xboole_0,k3_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(involutiveness_k24_polynom1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & l1_rlvect_1(B) & v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) ) => k24_polynom1(A,B,k24_polynom1(A,B,C)) = C ) ), file(polynom1,k24_polynom1), []). fof(l4_groeb_1,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & v2_bagorder(B,A) & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) ) => ! [C] : ( ( ~ v3_struct_0(C) & v2_group_1(C) & v4_group_1(C) & v7_group_1(C) & v4_rlvect_1(C) & v5_rlvect_1(C) & v6_rlvect_1(C) & v7_vectsp_1(C) & v9_vectsp_1(C) & ~ v10_vectsp_1(C) & l3_vectsp_1(C) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(D,k14_polynom1(A),C) & m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) ) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(E,k14_polynom1(A),C) & m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) ) => ! [F] : ( m1_subset_1(F,u1_struct_0(k30_polynom1(A,C))) => ! [G] : ( m1_subset_1(G,u1_struct_0(k30_polynom1(A,C))) => ( ( F = D & G = E ) => k6_rlvect_1(k30_polynom1(A,C),F,G) = k25_polynom1(A,C,D,E) ) ) ) ) ) ) ) ) ), file(groeb_1,l4_groeb_1), []). fof(rc10_polynom1,axiom,( ! [A] : ? [B] : ( m1_pboole(B,A) & v1_relat_1(B) & v1_funct_1(B) & v1_seq_1(B) ) ), file(polynom1,rc10_polynom1), []). fof(rc10_vectsp_1,axiom,( ? [A] : ( l3_vectsp_1(A) & ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v3_vectsp_1(A) & v4_vectsp_1(A) & v5_vectsp_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v9_vectsp_1(A) & ~ v10_vectsp_1(A) ) ), file(vectsp_1,rc10_vectsp_1), []). fof(rc11_polynom1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & ~ v1_xboole_0(A) & v1_seq_1(A) & v7_seqm_3(A) & v1_polynom1(A) ) ), file(polynom1,rc11_polynom1), []). fof(rc12_polynom1,axiom,( ! [A] : ? [B] : ( m1_relset_1(B,A,k5_numbers) & v1_relat_1(B) & v1_funct_1(B) & v1_funct_2(B,A,k5_numbers) & v1_seq_1(B) & v7_seqm_3(B) & v1_polynom1(B) ) ), file(polynom1,rc12_polynom1), []). fof(rc13_polynom1,axiom,( ! [A] : ? [B] : ( m1_pboole(B,A) & v1_relat_1(B) & v1_funct_1(B) & v1_seq_1(B) & v7_seqm_3(B) & v1_polynom1(B) ) ), file(polynom1,rc13_polynom1), []). fof(rc14_ideal_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l3_vectsp_1(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(B) & v1_ideal_1(B,A) & v2_ideal_1(B,A) & v3_ideal_1(B,A) & v6_ideal_1(B,A) ) ) ), file(ideal_1,rc14_ideal_1), []). fof(rc14_polynom1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & l2_struct_0(B) ) => ? [C] : ( m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) & v1_relat_1(C) & v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(C,k14_polynom1(A),B) ) ) ), file(polynom1,rc14_polynom1), []). fof(rc1_binom,axiom,( ? [A] : ( l1_rlvect_1(A) & ~ v3_struct_0(A) & v2_algstr_1(A) ) ), file(binom,rc1_binom), []). fof(rc1_card_1,axiom,( ? [A] : v1_card_1(A) ), file(card_1,rc1_card_1), []). fof(rc1_finset_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) ) ), file(finset_1,rc1_finset_1), []). fof(rc1_gcd_1,axiom,( ? [A] : ( l3_vectsp_1(A) & ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v3_vectsp_1(A) & v4_vectsp_1(A) & v5_vectsp_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v9_vectsp_1(A) & ~ v10_vectsp_1(A) & v2_vectsp_2(A) ) ), file(gcd_1,rc1_gcd_1), []). fof(rc1_ideal_1,axiom,( ? [A] : ( l1_rlvect_1(A) & ~ v3_struct_0(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v1_algstr_1(A) ) ), file(ideal_1,rc1_ideal_1), []). fof(rc1_int_1,axiom,( ? [A] : ( m1_subset_1(A,k1_numbers) & v1_xcmplx_0(A) & v1_xreal_0(A) & v1_int_1(A) ) ), file(int_1,rc1_int_1), []). fof(rc1_nat_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc1_nat_1), []). fof(rc1_ordinal1,axiom,( ? [A] : ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) ) ), file(ordinal1,rc1_ordinal1), []). fof(rc1_polynom2,axiom,( ? [A] : ( l3_vectsp_1(A) & ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v4_vectsp_1(A) & v5_vectsp_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v1_algstr_1(A) & v2_algstr_1(A) & v3_algstr_1(A) & v4_algstr_1(A) & v5_algstr_1(A) & v6_algstr_1(A) & ~ v3_realset2(A) ) ), file(polynom2,rc1_polynom2), []). fof(rc1_polynom4,axiom,( ? [A] : ( l3_vectsp_1(A) & ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v3_vectsp_1(A) & v4_vectsp_1(A) & v5_vectsp_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v9_vectsp_1(A) & ~ v10_vectsp_1(A) & v2_vectsp_2(A) & ~ v3_realset2(A) & v1_algstr_1(A) & v2_algstr_1(A) & v3_algstr_1(A) & v4_algstr_1(A) & v5_algstr_1(A) & v6_algstr_1(A) ) ), file(polynom4,rc1_polynom4), []). fof(rc1_polynom7,axiom,( ? [A] : ( l2_struct_0(A) & ~ v3_struct_0(A) & ~ v3_realset2(A) ) ), file(polynom7,rc1_polynom7), []). fof(rc1_polyred,axiom,( ! [A,B] : ( ( v3_ordinal1(A) & ~ v3_realset2(B) & l2_struct_0(B) ) => ? [C] : ( m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) & v1_relat_1(C) & v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(C,k14_polynom1(A),B) & v1_polynom7(C,A,B) & v3_polynom7(C,A,B) ) ) ), file(polyred,rc1_polyred), []). fof(rc1_relat_1,axiom,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) ) ), file(relat_1,rc1_relat_1), []). fof(rc1_rewrite1,axiom,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v2_relat_2(A) & v1_rewrite1(A) & v2_rewrite1(A) & v3_rewrite1(A) ) ), file(rewrite1,rc1_rewrite1), []). 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_termord,axiom,( ? [A] : ( l1_rlvect_1(A) & ~ v3_struct_0(A) & ~ v3_realset2(A) ) ), file(termord,rc1_termord), []). fof(rc2_binom,axiom,( ? [A] : ( l1_rlvect_1(A) & ~ v3_struct_0(A) & v3_algstr_1(A) ) ), file(binom,rc2_binom), []). fof(rc2_card_1,axiom,( ? [A] : ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v1_finset_1(A) & v1_card_1(A) ) ), file(card_1,rc2_card_1), []). 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_gcd_1,axiom,( ? [A] : ( l3_vectsp_1(A) & ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v4_vectsp_1(A) & v5_vectsp_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & ~ v10_vectsp_1(A) & v2_vectsp_2(A) & v3_gcd_1(A) ) ), file(gcd_1,rc2_gcd_1), []). fof(rc2_ideal_1,axiom,( ? [A] : ( l3_vectsp_1(A) & ~ v3_struct_0(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v4_vectsp_1(A) & v5_vectsp_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v1_binom(A) & v1_algstr_1(A) & v2_algstr_1(A) & v3_algstr_1(A) & ~ v3_realset2(A) ) ), file(ideal_1,rc2_ideal_1), []). fof(rc2_int_1,axiom,( ? [A] : v1_int_1(A) ), file(int_1,rc2_int_1), []). fof(rc2_nat_1,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(A) & v3_ordinal1(A) ) ), file(nat_1,rc2_nat_1), []). fof(rc2_ordinal1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) ) ), file(ordinal1,rc2_ordinal1), []). fof(rc2_polynom1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) & v1_funcop_1(A) & v1_matrlin(A) ) ), file(polynom1,rc2_polynom1), []). fof(rc2_polynom7,axiom,( ? [A] : ( l3_vectsp_1(A) & ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v4_vectsp_1(A) & v5_vectsp_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v1_algstr_1(A) & v2_algstr_1(A) & v3_algstr_1(A) & v4_algstr_1(A) & v5_algstr_1(A) & v6_algstr_1(A) & v2_vectsp_2(A) & v1_binom(A) & ~ v3_realset2(A) ) ), file(polynom7,rc2_polynom7), []). fof(rc2_polyred,axiom,( ? [A] : ( l3_vectsp_1(A) & ~ v3_struct_0(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v4_vectsp_1(A) & v5_vectsp_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v9_vectsp_1(A) & ~ v10_vectsp_1(A) & v2_vectsp_2(A) & v1_binom(A) & v1_algstr_1(A) & v2_algstr_1(A) & v3_algstr_1(A) & v4_algstr_1(A) & v5_algstr_1(A) & v6_algstr_1(A) & ~ v3_realset2(A) ) ), file(polyred,rc2_polyred), []). fof(rc2_relat_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) ) ), file(relat_1,rc2_relat_1), []). fof(rc2_rewrite1,axiom,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v2_relat_2(A) & v1_rewrite1(A) & v2_rewrite1(A) & v3_rewrite1(A) & v4_rewrite1(A) & v5_rewrite1(A) & v6_rewrite1(A) & v7_rewrite1(A) & v8_rewrite1(A) & v9_rewrite1(A) ) ), file(rewrite1,rc2_rewrite1), []). 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_termord,axiom,( ? [A] : ( l1_rlvect_1(A) & ~ v3_struct_0(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v1_algstr_1(A) & v2_algstr_1(A) & v3_algstr_1(A) & v4_algstr_1(A) & v5_algstr_1(A) & v6_algstr_1(A) & ~ v3_realset2(A) ) ), file(termord,rc2_termord), []). fof(rc3_binom,axiom,( ? [A] : ( l1_rlvect_1(A) & ~ v3_struct_0(A) & v1_binom(A) ) ), file(binom,rc3_binom), []). 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_gcd_1,axiom,( ? [A] : ( l1_vectsp_1(A) & ~ v3_struct_0(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v6_vectsp_1(A) & v8_vectsp_1(A) & v3_gcd_1(A) ) ), file(gcd_1,rc3_gcd_1), []). fof(rc3_ideal_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_rlvect_1(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(B) & v1_ideal_1(B,A) ) ) ), file(ideal_1,rc3_ideal_1), []). fof(rc3_nat_1,axiom,( ? [A] : ( m1_subset_1(A,k5_numbers) & ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc3_nat_1), []). fof(rc3_ordinal1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) ) ), file(ordinal1,rc3_ordinal1), []). fof(rc3_polynom7,axiom,( ! [A] : ( ( ~ v3_realset2(A) & l2_struct_0(A) ) => ? [B] : ( m1_subset_1(B,u1_struct_0(A)) & v8_rlvect_1(B,A) ) ) ), file(polynom7,rc3_polynom7), []). fof(rc3_relat_1,axiom,( ? [A] : ( v1_relat_1(A) & v3_relat_1(A) ) ), file(relat_1,rc3_relat_1), []). fof(rc3_rewrite1,axiom,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v2_relat_2(A) & v1_rewrite1(A) & v2_rewrite1(A) & v3_rewrite1(A) & v4_rewrite1(A) & v5_rewrite1(A) & v6_rewrite1(A) & v7_rewrite1(A) & v8_rewrite1(A) & v9_rewrite1(A) & v10_rewrite1(A) ) ), file(rewrite1,rc3_rewrite1), []). fof(rc3_struct_0,axiom,( ? [A] : ( l1_struct_0(A) & ~ v3_struct_0(A) ) ), file(struct_0,rc3_struct_0), []). fof(rc3_termord,axiom,( ! [A] : ( v3_ordinal1(A) => ? [B] : ( m1_relset_1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_1(B) & v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & v2_bagorder(B,A) ) ) ), file(termord,rc3_termord), []). fof(rc4_binom,axiom,( ? [A] : ( l3_vectsp_1(A) & ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v4_vectsp_1(A) & v5_vectsp_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v1_algstr_1(A) & v2_algstr_1(A) & v3_algstr_1(A) & v1_binom(A) ) ), file(binom,rc4_binom), []). 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_gcd_1,axiom,( ? [A] : ( l2_vectsp_1(A) & ~ v3_struct_0(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v6_vectsp_1(A) & v8_vectsp_1(A) & v3_gcd_1(A) ) ), file(gcd_1,rc4_gcd_1), []). fof(rc4_ideal_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_group_1(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(B) & v2_ideal_1(B,A) ) ) ), file(ideal_1,rc4_ideal_1), []). fof(rc4_polynom7,axiom,( ! [A,B] : ( ( ~ v3_realset2(B) & l2_struct_0(B) ) => ? [C] : ( m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) & v1_relat_1(C) & v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & v1_polynom7(C,A,B) ) ) ), file(polynom7,rc4_polynom7), []). fof(rc4_rewrite1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v2_relat_2(A) & v1_rewrite1(A) & v2_rewrite1(A) & v3_rewrite1(A) & v4_rewrite1(A) & v5_rewrite1(A) & v7_rewrite1(A) & v8_rewrite1(A) & v9_rewrite1(A) & v10_rewrite1(A) ) ), file(rewrite1,rc4_rewrite1), []). fof(rc4_struct_0,axiom,( ? [A] : ( l2_struct_0(A) & ~ v3_struct_0(A) ) ), file(struct_0,rc4_struct_0), []). fof(rc5_gcd_1,axiom,( ? [A] : ( l3_vectsp_1(A) & ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v4_vectsp_1(A) & v5_vectsp_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & ~ v10_vectsp_1(A) & v2_vectsp_2(A) & v3_gcd_1(A) ) ), file(gcd_1,rc5_gcd_1), []). fof(rc5_ideal_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_group_1(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(B) & v3_ideal_1(B,A) ) ) ), file(ideal_1,rc5_ideal_1), []). fof(rc5_polynom7,axiom,( ! [A,B] : ( ( v3_ordinal1(A) & ~ v3_realset2(B) & l2_struct_0(B) ) => ? [C] : ( m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) & v1_relat_1(C) & v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(C,k14_polynom1(A),B) & v1_polynom7(C,A,B) ) ) ), file(polynom7,rc5_polynom7), []). 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_ideal_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l3_vectsp_1(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(B) & v1_ideal_1(B,A) & v2_ideal_1(B,A) & v3_ideal_1(B,A) ) ) ), file(ideal_1,rc6_ideal_1), []). fof(rc6_polynom1,axiom,( ? [A] : ( l3_vectsp_1(A) & ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v3_vectsp_1(A) & v4_vectsp_1(A) & v5_vectsp_1(A) & v6_vectsp_1(A) & v7_vectsp_1(A) & v8_vectsp_1(A) & v9_vectsp_1(A) & ~ v3_realset2(A) & v1_algstr_1(A) & v2_algstr_1(A) & v3_algstr_1(A) & v4_algstr_1(A) & v5_algstr_1(A) & v6_algstr_1(A) ) ), file(polynom1,rc6_polynom1), []). fof(rc6_vectsp_1,axiom,( ? [A] : ( l3_vectsp_1(A) & v3_vectsp_1(A) ) ), file(vectsp_1,rc6_vectsp_1), []). fof(rc7_ideal_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l3_vectsp_1(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(B) & v1_ideal_1(B,A) & v3_ideal_1(B,A) ) ) ), file(ideal_1,rc7_ideal_1), []). fof(rc7_polynom1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) & v7_seqm_3(A) ) ), file(polynom1,rc7_polynom1), []). fof(rc7_polynom7,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & l2_struct_0(B) ) => ? [C] : ( m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) & v1_relat_1(C) & v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & v3_polynom7(C,A,B) ) ) ), file(polynom7,rc7_polynom7), []). fof(rc7_vectsp_1,axiom,( ? [A] : ( l3_vectsp_1(A) & ~ v3_struct_0(A) & v3_vectsp_1(A) ) ), file(vectsp_1,rc7_vectsp_1), []). fof(rc8_ideal_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l3_vectsp_1(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(B) & v1_ideal_1(B,A) & v2_ideal_1(B,A) ) ) ), file(ideal_1,rc8_ideal_1), []). fof(rc8_polynom1,axiom,( ! [A] : ? [B] : ( m1_pboole(B,A) & v1_relat_1(B) & v1_funct_1(B) & v1_seq_1(B) & v7_seqm_3(B) ) ), file(polynom1,rc8_polynom1), []). fof(rc8_polynom7,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & l2_struct_0(B) ) => ? [C] : ( m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) & v1_relat_1(C) & v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & v4_polynom7(C,A,B) ) ) ), file(polynom7,rc8_polynom7), []). fof(rc8_vectsp_1,axiom,( ? [A] : ( l1_group_1(A) & ~ v3_struct_0(A) & v4_group_1(A) & v7_group_1(A) ) ), file(vectsp_1,rc8_vectsp_1), []). fof(rc9_polynom1,axiom,( ! [A] : ? [B] : ( m1_pboole(B,A) & v1_relat_1(B) & v1_funct_1(B) & v1_seq_1(B) ) ), file(polynom1,rc9_polynom1), []). fof(rc9_vectsp_1,axiom,( ? [A] : ( l1_vectsp_1(A) & ~ v3_struct_0(A) & v2_group_1(A) & v4_group_1(A) & v7_group_1(A) & v6_vectsp_1(A) & v8_vectsp_1(A) ) ), file(vectsp_1,rc9_vectsp_1), []). fof(redefinition_k13_ideal_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) ) => k13_ideal_1(A,B,C) = k3_xboole_0(B,C) ) ), file(ideal_1,k13_ideal_1), []). fof(redefinition_k14_polynom1,axiom,( ! [A] : k14_polynom1(A) = k13_polynom1(A) ), file(polynom1,k14_polynom1), []). fof(redefinition_k15_polynom1,axiom,( ! [A,B,C,D] : ( ( ~ v3_struct_0(B) & l1_struct_0(B) & v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) & v7_seqm_3(D) & v1_polynom1(D) & m1_pboole(D,A) ) => k15_polynom1(A,B,C,D) = k1_funct_1(C,D) ) ), file(polynom1,k15_polynom1), []). fof(redefinition_k1_groeb_1,axiom,( ! [A,B,C] : ( ( v3_ordinal1(A) & v2_group_1(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v7_vectsp_1(B) & ~ v3_realset2(B) & l3_vectsp_1(B) & v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(C,k14_polynom1(A),B) & m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) ) => k1_groeb_1(A,B,C) = k1_tarski(C) ) ), file(groeb_1,k1_groeb_1), []). fof(redefinition_k23_polynom1,axiom,( ! [A,B,C,D] : ( ( ~ v3_struct_0(B) & v3_rlvect_1(B) & v5_rlvect_1(B) & l1_rlvect_1(B) & v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) & v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(B)) & m1_relset_1(D,k14_polynom1(A),u1_struct_0(B)) ) => k23_polynom1(A,B,C,D) = k22_polynom1(A,B,C,D) ) ), file(polynom1,k23_polynom1), []). fof(redefinition_k4_rlvect_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & l1_rlvect_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => k4_rlvect_1(A,B,C) = k2_rlvect_1(A,B,C) ) ), file(rlvect_1,k4_rlvect_1), []). fof(redefinition_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_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_k8_funct_2,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & m1_subset_1(D,A) ) => k8_funct_2(A,B,C,D) = k1_funct_1(C,D) ) ), file(funct_2,k8_funct_2), []). fof(redefinition_m1_polynom1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(k14_polynom1(A))) ) => ! [C] : ( m1_polynom1(C,A,B) <=> m1_subset_1(C,B) ) ) ), file(polynom1,m1_polynom1), []). 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_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_r3_polynom1,axiom,( ! [A,B,C] : ( ( v7_seqm_3(B) & v1_polynom1(B) & m1_pboole(B,A) & v7_seqm_3(C) & v1_polynom1(C) & m1_pboole(C,A) ) => r3_polynom1(A,B,B) ) ), file(polynom1,r3_polynom1), []). 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(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_termord,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_2(B) & v4_relat_2(B) & v8_relat_2(B) & v2_bagorder(B,A) & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) ) => ! [C] : ( ( v7_seqm_3(C) & v1_polynom1(C) & m1_pboole(C,A) ) => ! [D] : ( ( v7_seqm_3(D) & v1_polynom1(D) & m1_pboole(D,A) ) => ( r3_polynom1(A,C,D) => r1_termord(A,B,C,D) ) ) ) ) ) ), file(termord,t10_termord), []). fof(t12_groeb_1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & v2_bagorder(B,A) & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) ) => ! [C] : ( ( ~ v3_struct_0(C) & v2_group_1(C) & v4_group_1(C) & v7_group_1(C) & v3_rlvect_1(C) & v4_rlvect_1(C) & v5_rlvect_1(C) & v6_rlvect_1(C) & v7_vectsp_1(C) & v9_vectsp_1(C) & ~ v10_vectsp_1(C) & l3_vectsp_1(C) ) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) => ( v9_rewrite1(k3_polyred(A,B,C,D)) => v7_rewrite1(k3_polyred(A,B,C,D)) ) ) ) ) ) ), file(groeb_1,t12_groeb_1), []). fof(t12_termord,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_2(B) & v4_relat_2(B) & v8_relat_2(B) & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) ) => ! [C] : ( ( v7_seqm_3(C) & v1_polynom1(C) & m1_pboole(C,A) ) => ! [D] : ( ( v7_seqm_3(D) & v1_polynom1(D) & m1_pboole(D,A) ) => ( r6_pboole(A,k2_termord(A,B,C,D),C) | r6_pboole(A,k2_termord(A,B,C,D),D) ) ) ) ) ) ), file(termord,t12_termord), []). fof(t13_groeb_1,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) ) => ! [C] : ( ( v2_group_1(C) & v4_group_1(C) & v7_group_1(C) & v4_rlvect_1(C) & v5_rlvect_1(C) & v6_rlvect_1(C) & v7_vectsp_1(C) & v9_vectsp_1(C) & ~ v3_realset2(C) & l3_vectsp_1(C) ) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) => ( v7_rewrite1(k3_polyred(A,B,C,D)) => v4_rewrite1(k3_polyred(A,B,C,D)) ) ) ) ) ) ), file(groeb_1,t13_groeb_1), []). fof(t14_groeb_1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & v2_bagorder(B,A) & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) ) => ! [C] : ( ( ~ v3_struct_0(C) & v2_group_1(C) & v4_group_1(C) & v7_group_1(C) & v3_rlvect_1(C) & v4_rlvect_1(C) & v5_rlvect_1(C) & v6_rlvect_1(C) & v7_vectsp_1(C) & v9_vectsp_1(C) & ~ v10_vectsp_1(C) & l3_vectsp_1(C) ) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) => ( v4_rewrite1(k3_polyred(A,B,C,D)) => v8_rewrite1(k3_polyred(A,B,C,D)) ) ) ) ) ) ), file(groeb_1,t14_groeb_1), []). fof(t15_groeb_1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & v2_bagorder(B,A) & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) ) => ! [C] : ( ( ~ v3_struct_0(C) & v2_group_1(C) & v4_group_1(C) & v7_group_1(C) & v3_rlvect_1(C) & v4_rlvect_1(C) & v5_rlvect_1(C) & v6_rlvect_1(C) & v7_vectsp_1(C) & v9_vectsp_1(C) & ~ v10_vectsp_1(C) & l3_vectsp_1(C) ) => ! [D] : ( ( ~ v1_xboole_0(D) & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) ) => ( v8_rewrite1(k3_polyred(A,B,C,D)) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(E,k14_polynom1(A),C) & m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) ) => ( r2_hidden(E,k7_ideal_1(k30_polynom1(A,C),D)) => r1_rewrite1(k3_polyred(A,B,C,D),E,k26_polynom1(A,C)) ) ) ) ) ) ) ) ), file(groeb_1,t15_groeb_1), []). fof(t15_ideal_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v5_vectsp_1(A) & v8_vectsp_1(A) & l3_vectsp_1(A) ) => ! [B] : ( ( ~ v1_xboole_0(B) & v1_ideal_1(B,A) & v2_ideal_1(B,A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ( ( r2_hidden(C,B) & r2_hidden(D,B) ) => r2_hidden(k6_rlvect_1(A,C,D),B) ) ) ) ) ) ), file(ideal_1,t15_ideal_1), []). fof(t16_groeb_1,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) ) => ! [C] : ( ( v2_group_1(C) & v4_group_1(C) & v7_group_1(C) & v4_rlvect_1(C) & v5_rlvect_1(C) & v6_rlvect_1(C) & v7_vectsp_1(C) & v9_vectsp_1(C) & ~ v3_realset2(C) & l3_vectsp_1(C) ) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) => ( ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(E,k14_polynom1(A),C) & m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) ) => ( r2_hidden(E,k7_ideal_1(k30_polynom1(A,C),D)) => r1_rewrite1(k3_polyred(A,B,C,D),E,k26_polynom1(A,C)) ) ) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(E,k14_polynom1(A),C) & v1_polynom7(E,A,C) & m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) ) => ( r2_hidden(E,k7_ideal_1(k30_polynom1(A,C),D)) => r7_polyred(A,B,C,E,D) ) ) ) ) ) ) ) ), file(groeb_1,t16_groeb_1), []). fof(t16_rlvect_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & l1_rlvect_1(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ( k2_rlvect_1(A,B,k5_rlvect_1(A,B)) = k1_rlvect_1(A) & k2_rlvect_1(A,k5_rlvect_1(A,B),B) = k1_rlvect_1(A) ) ) ) ), file(rlvect_1,t16_rlvect_1), []). fof(t17_groeb_1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & v2_bagorder(B,A) & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) ) => ! [C] : ( ( ~ v3_struct_0(C) & v2_group_1(C) & v4_group_1(C) & v7_group_1(C) & v3_rlvect_1(C) & v4_rlvect_1(C) & v5_rlvect_1(C) & v6_rlvect_1(C) & v7_vectsp_1(C) & v9_vectsp_1(C) & ~ v10_vectsp_1(C) & l3_vectsp_1(C) ) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) => ( ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(E,k14_polynom1(A),C) & v1_polynom7(E,A,C) & m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) ) => ( r2_hidden(E,k7_ideal_1(k30_polynom1(A,C),D)) => r7_polyred(A,B,C,E,D) ) ) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(E,k14_polynom1(A),C) & v1_polynom7(E,A,C) & m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) ) => ( r2_hidden(E,k7_ideal_1(k30_polynom1(A,C),D)) => r10_polyred(A,B,C,E,D) ) ) ) ) ) ) ) ), file(groeb_1,t17_groeb_1), []). fof(t17_termord,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) ) => ! [C] : ( ( ~ v3_struct_0(C) & l2_struct_0(C) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(D,k14_polynom1(A),C) & m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) ) => ( k4_termord(A,B,C,D) = k1_rlvect_1(C) <=> D = k26_polynom1(A,C) ) ) ) ) ) ), file(termord,t17_termord), []). fof(t18_groeb_1,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) ) => ! [C] : ( ( v2_group_1(C) & v4_group_1(C) & v7_group_1(C) & v4_rlvect_1(C) & v5_rlvect_1(C) & v6_rlvect_1(C) & v7_vectsp_1(C) & v9_vectsp_1(C) & ~ v3_realset2(C) & l3_vectsp_1(C) ) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) => ( ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(E,k14_polynom1(A),C) & v1_polynom7(E,A,C) & m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) ) => ( r2_hidden(E,k7_ideal_1(k30_polynom1(A,C),D)) => r10_polyred(A,B,C,E,D) ) ) => ! [E] : ( ( v7_seqm_3(E) & v1_polynom1(E) & m1_pboole(E,A) ) => ~ ( r2_hidden(E,k2_groeb_1(A,B,C,k7_ideal_1(k30_polynom1(A,C),D))) & ! [F] : ( ( v7_seqm_3(F) & v1_polynom1(F) & m1_pboole(F,A) ) => ~ ( r2_hidden(F,k2_groeb_1(A,B,C,D)) & r3_polynom1(A,F,E) ) ) ) ) ) ) ) ) ) ), file(groeb_1,t18_groeb_1), []). fof(t1_boole,axiom,( ! [A] : k2_xboole_0(A,k1_xboole_0) = A ), file(boole,t1_boole), []). fof(t1_polynom7,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & l2_struct_0(B) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) ) => ( C = k26_polynom1(A,B) <=> k12_polynom1(k14_polynom1(A),B,C) = k1_xboole_0 ) ) ) ), file(polynom7,t1_polynom7), []). fof(t1_subset,axiom,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), []). fof(t29_polyred,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( ( v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) ) => ! [C] : ( ( ~ v3_struct_0(C) & l1_rlvect_1(C) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(D,k14_polynom1(A),C) & m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) ) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(E,k14_polynom1(A),C) & m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) ) => ( r1_polyred(A,B,C,D,E) <=> ~ r2_polyred(A,B,C,E,D) ) ) ) ) ) ) ), file(polyred,t29_polyred), []). fof(t2_boole,axiom,( ! [A] : k3_xboole_0(A,k1_xboole_0) = k1_xboole_0 ), file(boole,t2_boole), []). fof(t2_polyred,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & v1_algstr_1(B) & l1_rlvect_1(B) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) ) => k22_polynom1(A,B,k26_polynom1(A,B),C) = C ) ) ), file(polyred,t2_polyred), []). 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(t31_polyred,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v2_bagorder(B,A) & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) ) => ! [C] : ( ( v2_group_1(C) & v4_rlvect_1(C) & v5_rlvect_1(C) & v6_rlvect_1(C) & v7_vectsp_1(C) & ~ v3_realset2(C) & l3_vectsp_1(C) ) => ! [D] : ( ( ~ v1_xboole_0(D) & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) ) => ? [E] : ( v1_funct_1(E) & v1_funct_2(E,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(E,k14_polynom1(A),C) & m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) & r2_hidden(E,D) & ! [F] : ( ( v1_funct_1(F) & v1_funct_2(F,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(F,k14_polynom1(A),C) & m2_relset_1(F,k14_polynom1(A),u1_struct_0(C)) ) => ( r2_hidden(F,D) => r1_polyred(A,B,C,E,F) ) ) ) ) ) ) ) ), file(polyred,t31_polyred), []). fof(t32_polyred,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( ( v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v2_bagorder(B,A) & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) ) => ! [C] : ( ( v4_rlvect_1(C) & v5_rlvect_1(C) & v6_rlvect_1(C) & ~ v3_realset2(C) & l1_rlvect_1(C) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(D,k14_polynom1(A),C) & m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) ) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(E,k14_polynom1(A),C) & m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) ) => ( r2_polyred(A,B,C,D,E) <=> ~ ( ~ ( D = k26_polynom1(A,C) & E != k26_polynom1(A,C) ) & ~ r2_termord(A,B,k3_termord(A,B,C,D),k3_termord(A,B,C,E)) & ~ ( k3_termord(A,B,C,D) = k3_termord(A,B,C,E) & r2_polyred(A,B,C,k6_termord(A,B,C,D),k6_termord(A,B,C,E)) ) ) ) ) ) ) ) ) ), file(polyred,t32_polyred), []). fof(t36_polyred,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( ( v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) ) => ! [C] : ( ( v2_group_1(C) & v4_group_1(C) & v7_group_1(C) & v4_rlvect_1(C) & v5_rlvect_1(C) & v6_rlvect_1(C) & v7_vectsp_1(C) & v9_vectsp_1(C) & ~ v3_realset2(C) & l3_vectsp_1(C) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(D,k14_polynom1(A),C) & m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) ) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(E,k14_polynom1(A),C) & v1_polynom7(E,A,C) & m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) ) => ( r6_polyred(A,B,C,D,E) <=> ? [F] : ( v7_seqm_3(F) & v1_polynom1(F) & m1_pboole(F,A) & r2_hidden(F,k12_polynom1(k14_polynom1(A),C,D)) & r3_polynom1(A,k3_termord(A,B,C,E),F) ) ) ) ) ) ) ) ), file(polyred,t36_polyred), []). fof(t37_xboole_1,axiom,( ! [A,B] : ( k4_xboole_0(A,B) = k1_xboole_0 <=> r1_tarski(A,B) ) ), file(xboole_1,t37_xboole_1), []). fof(t3_boole,axiom,( ! [A] : k4_xboole_0(A,k1_xboole_0) = A ), file(boole,t3_boole), []). fof(t3_polyred,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & l1_rlvect_1(B) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) ) => ( k22_polynom1(A,B,k24_polynom1(A,B,C),C) = k26_polynom1(A,B) & k22_polynom1(A,B,C,k24_polynom1(A,B,C)) = k26_polynom1(A,B) ) ) ) ), file(polyred,t3_polyred), []). fof(t3_subset,axiom,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), []). fof(t4_boole,axiom,( ! [A] : k4_xboole_0(k1_xboole_0,A) = k1_xboole_0 ), file(boole,t4_boole), []). 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_groeb_1,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( ( ~ v3_struct_0(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & l3_vectsp_1(B) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & v2_polynom1(C,k14_polynom1(A),B) & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) ) => k12_polynom1(k14_polynom1(A),B,k24_polynom1(A,B,C)) = k12_polynom1(k14_polynom1(A),B,C) ) ) ) ), file(groeb_1,t5_groeb_1), []). 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(t5_termord,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) ) => ! [C] : ( ( v7_seqm_3(C) & v1_polynom1(C) & m1_pboole(C,A) ) => ! [D] : ( ( v7_seqm_3(D) & v1_polynom1(D) & m1_pboole(D,A) ) => ( r1_termord(A,B,C,D) <=> ~ r2_termord(A,B,D,C) ) ) ) ) ) ), file(termord,t5_termord), []). fof(t6_boole,axiom,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), []). fof(t79_polynom1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & v5_rlvect_1(B) & l1_rlvect_1(B) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(B)) & m2_relset_1(D,k14_polynom1(A),u1_struct_0(B)) ) => r1_tarski(k12_polynom1(k14_polynom1(A),B,k22_polynom1(A,B,C,D)),k4_subset_1(k14_polynom1(A),k12_polynom1(k14_polynom1(A),B,C),k12_polynom1(k14_polynom1(A),B,D))) ) ) ) ), file(polynom1,t79_polynom1), []). fof(t7_boole,axiom,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), []). fof(t7_groeb_1,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & v2_bagorder(B,A) & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) ) => ! [C] : ( ( ~ v3_struct_0(C) & v2_group_1(C) & v4_rlvect_1(C) & v5_rlvect_1(C) & v6_rlvect_1(C) & v7_vectsp_1(C) & ~ v3_realset2(C) & l3_vectsp_1(C) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(D,k14_polynom1(A),C) & m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) ) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(E,k14_polynom1(A),C) & m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) ) => r1_termord(A,B,k3_termord(A,B,C,k25_polynom1(A,C,D,E)),k2_termord(A,B,k3_termord(A,B,C,D),k3_termord(A,B,C,E))) ) ) ) ) ) ), file(groeb_1,t7_groeb_1), []). fof(t7_termord,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_2(B) & v4_relat_2(B) & v8_relat_2(B) & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) ) => ! [C] : ( ( v7_seqm_3(C) & v1_polynom1(C) & m1_pboole(C,A) ) => ! [D] : ( ( v7_seqm_3(D) & v1_polynom1(D) & m1_pboole(D,A) ) => ( ( r1_termord(A,B,C,D) & r1_termord(A,B,D,C) ) => r6_pboole(A,C,D) ) ) ) ) ) ), file(termord,t7_termord), []). fof(t80_polynom1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & l3_vectsp_1(B) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(B)) & m2_relset_1(D,k14_polynom1(A),u1_struct_0(B)) ) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,k14_polynom1(A),u1_struct_0(B)) & m2_relset_1(E,k14_polynom1(A),u1_struct_0(B)) ) => k22_polynom1(A,B,k22_polynom1(A,B,C,D),E) = k22_polynom1(A,B,C,k22_polynom1(A,B,D,E)) ) ) ) ) ), file(polynom1,t80_polynom1), []). fof(t82_polynom1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & v5_rlvect_1(B) & l1_rlvect_1(B) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B)) & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) ) => k22_polynom1(A,B,C,k26_polynom1(A,B)) = C ) ) ), file(polynom1,t82_polynom1), []). fof(t8_boole,axiom,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), []). fof(t9_groeb_1,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A)) & v1_relat_2(B) & v4_relat_2(B) & v6_relat_2(B) & v8_relat_2(B) & v2_bagorder(B,A) & m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) ) => ! [C] : ( ( ~ v3_struct_0(C) & v2_group_1(C) & v4_group_1(C) & v7_group_1(C) & v4_rlvect_1(C) & v5_rlvect_1(C) & v6_rlvect_1(C) & v7_vectsp_1(C) & v9_vectsp_1(C) & ~ v10_vectsp_1(C) & l3_vectsp_1(C) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(D,k14_polynom1(A),C) & v1_polynom7(D,A,C) & m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) ) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,k14_polynom1(A),u1_struct_0(C)) & v2_polynom1(E,k14_polynom1(A),C) & v1_polynom7(E,A,C) & m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) ) => ( r6_polyred(A,B,C,D,E) => r1_termord(A,B,k3_termord(A,B,C,E),k3_termord(A,B,C,D)) ) ) ) ) ) ) ), file(groeb_1,t9_groeb_1), []).