% Mizar problem: t49_turing_1,turing_1,2800,41 fof(t49_turing_1,conjecture,( ! [A] : ( l1_turing_1(A) => ! [B] : ( l1_turing_1(B) => ! [C] : ( m1_subset_1(C,k3_zfmisc_1(u2_turing_1(B),u1_turing_1(B),k8_domain_1(k1_numbers,k1_real_1(1),0,1))) => ! [D] : ( m1_subset_1(D,u2_turing_1(B)) => ! [E] : ( m1_subset_1(E,u1_turing_1(B)) => ( C = k8_funct_2(k2_zfmisc_1(u2_turing_1(B),u1_turing_1(B)),k3_zfmisc_1(u2_turing_1(B),u1_turing_1(B),k8_domain_1(k1_numbers,k1_real_1(1),0,1)),u3_turing_1(B),k1_domain_1(u2_turing_1(B),u1_turing_1(B),D,E)) => k1_funct_1(u3_turing_1(k32_turing_1(A,B)),k1_domain_1(k2_zfmisc_1(u2_turing_1(A),u2_turing_1(B)),u1_turing_1(B),k1_domain_1(u2_turing_1(A),u2_turing_1(B),u5_turing_1(A),D),E)) = k4_domain_1(k2_zfmisc_1(u2_turing_1(A),u2_turing_1(B)),u1_turing_1(B),k8_domain_1(k1_numbers,k1_real_1(1),0,1),k1_domain_1(u2_turing_1(A),u2_turing_1(B),u5_turing_1(A),k5_mcart_1(u2_turing_1(B),u1_turing_1(B),k8_domain_1(k1_numbers,k1_real_1(1),0,1),C)),k6_mcart_1(u2_turing_1(B),u1_turing_1(B),k8_domain_1(k1_numbers,k1_real_1(1),0,1),C),k7_mcart_1(u2_turing_1(B),u1_turing_1(B),k8_domain_1(k1_numbers,k1_real_1(1),0,1),C)) ) ) ) ) ) ) ), inference(mizar_bg_added,[status(thm)],[abstractness_v1_turing_1,antisymmetry_r2_hidden,cc1_arytm_3,cc1_finset_1,cc1_funct_1,cc1_relset_1,cc2_arytm_3,cc2_finset_1,cc2_funct_1,cc2_int_1,cc3_arytm_3,cc3_int_1,cc4_int_1,commutativity_k2_tarski,commutativity_k2_xboole_0,d1_mcart_1,d1_tarski,d23_turing_1,d24_turing_1,d25_turing_1,d26_turing_1,d27_turing_1,d29_turing_1,d2_mcart_1,d2_xboole_0,d2_zfmisc_1,d30_turing_1,d31_turing_1,d32_turing_1,d5_tarski,dt_g1_turing_1,dt_k1_domain_1,dt_k1_enumset1,dt_k1_funct_1,dt_k1_mcart_1,dt_k1_numbers,dt_k1_real_1,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k21_turing_1,dt_k22_turing_1,dt_k23_turing_1,dt_k24_turing_1,dt_k25_turing_1,dt_k26_turing_1,dt_k27_turing_1,dt_k28_turing_1,dt_k29_turing_1,dt_k2_domain_1,dt_k2_mcart_1,dt_k2_tarski,dt_k2_xboole_0,dt_k2_zfmisc_1,dt_k30_turing_1,dt_k31_turing_1,dt_k32_turing_1,dt_k3_domain_1,dt_k3_mcart_1,dt_k3_zfmisc_1,dt_k4_domain_1,dt_k4_tarski,dt_k4_xcmplx_0,dt_k5_mcart_1,dt_k5_numbers,dt_k5_ordinal2,dt_k6_domain_1,dt_k6_mcart_1,dt_k7_mcart_1,dt_k8_domain_1,dt_k8_funct_2,dt_l1_turing_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_u1_turing_1,dt_u2_turing_1,dt_u3_turing_1,dt_u4_turing_1,dt_u5_turing_1,existence_l1_turing_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,fc14_finset_1,fc15_finset_1,fc1_finset_1,fc1_numbers,fc1_ordinal2,fc1_subset_1,fc1_xboole_0,fc2_comput_1,fc2_finset_1,fc2_mod_2,fc2_subset_1,fc2_xboole_0,fc3_finset_1,fc3_int_1,fc3_subset_1,fc3_xboole_0,fc4_subset_1,fc5_comput_1,fc5_int_1,fc5_subset_1,fc9_finset_1,free_g1_turing_1,idempotence_k2_xboole_0,involutiveness_k1_real_1,involutiveness_k4_xcmplx_0,rc1_arytm_3,rc1_finset_1,rc1_funct_1,rc1_int_1,rc1_subset_1,rc1_turing_1,rc1_xboole_0,rc2_finset_1,rc2_funct_1,rc2_int_1,rc2_subset_1,rc2_xboole_0,rc3_finset_1,rc3_funct_1,rc4_comput_1,rc4_finset_1,rc4_funct_1,rc9_comput_1,redefinition_k1_domain_1,redefinition_k1_real_1,redefinition_k24_turing_1,redefinition_k25_turing_1,redefinition_k2_domain_1,redefinition_k3_domain_1,redefinition_k4_domain_1,redefinition_k5_numbers,redefinition_k6_domain_1,redefinition_k8_domain_1,redefinition_k8_funct_2,redefinition_m2_relset_1,reflexivity_r1_tarski,spc0_boole,spc1_boole,spc1_numerals,t1_boole,t1_numerals,t1_subset,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole]), [file(turing_1,t49_turing_1)]). fof(abstractness_v1_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => ( v1_turing_1(A) => A = g1_turing_1(u1_turing_1(A),u2_turing_1(A),u3_turing_1(A),u4_turing_1(A),u5_turing_1(A)) ) ) ), file(turing_1,v1_turing_1), []). fof(antisymmetry_r2_hidden,axiom,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), []). fof(cc1_arytm_3,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_ordinal1(B) & v2_ordinal1(B) & v3_ordinal1(B) ) ) ) ), file(arytm_3,cc1_arytm_3), []). fof(cc1_finset_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_finset_1(A) ) ), file(finset_1,cc1_finset_1), []). fof(cc1_funct_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_funct_1(A) ) ), file(funct_1,cc1_funct_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(cc2_arytm_3,axiom,( ! [A] : ( ( v1_xboole_0(A) & v3_ordinal1(A) ) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ) ), file(arytm_3,cc2_arytm_3), []). fof(cc2_finset_1,axiom,( ! [A] : ( v1_finset_1(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_finset_1(B) ) ) ), file(finset_1,cc2_finset_1), []). fof(cc2_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ) ), file(funct_1,cc2_funct_1), []). fof(cc2_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(cc3_arytm_3,axiom,( ! [A] : ( m1_subset_1(A,k5_ordinal2) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ) ), file(arytm_3,cc3_arytm_3), []). fof(cc3_int_1,axiom,( ! [A] : ( v4_ordinal2(A) => v1_int_1(A) ) ), file(int_1,cc3_int_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(commutativity_k2_tarski,axiom,( ! [A,B] : k2_tarski(A,B) = k2_tarski(B,A) ), file(tarski,k2_tarski), []). fof(commutativity_k2_xboole_0,axiom,( ! [A,B] : k2_xboole_0(A,B) = k2_xboole_0(B,A) ), file(xboole_0,k2_xboole_0), []). fof(d1_mcart_1,axiom,( ! [A] : ( ? [B,C] : A = k4_tarski(B,C) => ! [B] : ( B = k1_mcart_1(A) <=> ! [C,D] : ( A = k4_tarski(C,D) => B = C ) ) ) ), file(mcart_1,d1_mcart_1), []). fof(d1_tarski,axiom,( ! [A,B] : ( B = k1_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> C = A ) ) ), file(tarski,d1_tarski), []). fof(d23_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => ! [B] : ( l1_turing_1(B) => k21_turing_1(A,B) = k2_xboole_0(k2_zfmisc_1(u2_turing_1(A),k6_domain_1(u2_turing_1(B),u4_turing_1(B))),k2_zfmisc_1(k6_domain_1(u2_turing_1(A),u5_turing_1(A)),u2_turing_1(B))) ) ) ), file(turing_1,d23_turing_1), []). fof(d24_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => ! [B] : ( l1_turing_1(B) => ! [C] : ( m1_subset_1(C,k3_zfmisc_1(u2_turing_1(A),u1_turing_1(A),k8_domain_1(k1_numbers,k1_real_1(1),0,1))) => k22_turing_1(A,B,C) = k4_domain_1(k2_zfmisc_1(u2_turing_1(A),u2_turing_1(B)),u1_turing_1(A),k8_domain_1(k1_numbers,k1_real_1(1),0,1),k1_domain_1(u2_turing_1(A),u2_turing_1(B),k5_mcart_1(u2_turing_1(A),u1_turing_1(A),k8_domain_1(k1_numbers,k1_real_1(1),0,1),C),u4_turing_1(B)),k6_mcart_1(u2_turing_1(A),u1_turing_1(A),k8_domain_1(k1_numbers,k1_real_1(1),0,1),C),k7_mcart_1(u2_turing_1(A),u1_turing_1(A),k8_domain_1(k1_numbers,k1_real_1(1),0,1),C)) ) ) ) ), file(turing_1,d24_turing_1), []). fof(d25_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => ! [B] : ( l1_turing_1(B) => ! [C] : ( m1_subset_1(C,k3_zfmisc_1(u2_turing_1(B),u1_turing_1(B),k8_domain_1(k1_numbers,k1_real_1(1),0,1))) => k23_turing_1(A,B,C) = k4_domain_1(k2_zfmisc_1(u2_turing_1(A),u2_turing_1(B)),u1_turing_1(B),k8_domain_1(k1_numbers,k1_real_1(1),0,1),k1_domain_1(u2_turing_1(A),u2_turing_1(B),u5_turing_1(A),k5_mcart_1(u2_turing_1(B),u1_turing_1(B),k8_domain_1(k1_numbers,k1_real_1(1),0,1),C)),k6_mcart_1(u2_turing_1(B),u1_turing_1(B),k8_domain_1(k1_numbers,k1_real_1(1),0,1),C),k7_mcart_1(u2_turing_1(B),u1_turing_1(B),k8_domain_1(k1_numbers,k1_real_1(1),0,1),C)) ) ) ) ), file(turing_1,d25_turing_1), []). fof(d26_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => ! [B] : ( l1_turing_1(B) => ! [C] : ( m1_subset_1(C,k2_zfmisc_1(k21_turing_1(A,B),k2_xboole_0(u1_turing_1(A),u1_turing_1(B)))) => k26_turing_1(A,B,C) = k24_turing_1(A,B,k2_domain_1(k21_turing_1(A,B),k2_xboole_0(u1_turing_1(A),u1_turing_1(B)),C)) ) ) ) ), file(turing_1,d26_turing_1), []). fof(d27_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => ! [B] : ( l1_turing_1(B) => ! [C] : ( m1_subset_1(C,k2_zfmisc_1(k21_turing_1(A,B),k2_xboole_0(u1_turing_1(A),u1_turing_1(B)))) => k27_turing_1(A,B,C) = k25_turing_1(A,B,k2_domain_1(k21_turing_1(A,B),k2_xboole_0(u1_turing_1(A),u1_turing_1(B)),C)) ) ) ) ), file(turing_1,d27_turing_1), []). fof(d29_turing_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( ~ v1_xboole_0(C) => ! [D] : ( m1_subset_1(D,k2_zfmisc_1(A,k2_xboole_0(B,C))) => ( ? [E,F] : ( m1_subset_1(F,C) & D = k4_tarski(E,F) ) => k29_turing_1(A,B,C,D) = k3_domain_1(A,k2_xboole_0(B,C),D) ) ) ) ) ) ), file(turing_1,d29_turing_1), []). fof(d2_mcart_1,axiom,( ! [A] : ( ? [B,C] : A = k4_tarski(B,C) => ! [B] : ( B = k2_mcart_1(A) <=> ! [C,D] : ( A = k4_tarski(C,D) => B = D ) ) ) ), file(mcart_1,d2_mcart_1), []). 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(d2_zfmisc_1,axiom,( ! [A,B,C] : ( C = k2_zfmisc_1(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ? [E,F] : ( r2_hidden(E,A) & r2_hidden(F,B) & D = k4_tarski(E,F) ) ) ) ), file(zfmisc_1,d2_zfmisc_1), []). fof(d30_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => ! [B] : ( l1_turing_1(B) => ! [C] : ( m1_subset_1(C,k2_zfmisc_1(k21_turing_1(A,B),k2_xboole_0(u1_turing_1(A),u1_turing_1(B)))) => ( ( ? [D] : ( m1_subset_1(D,u2_turing_1(A)) & ? [E] : ( m1_subset_1(E,u1_turing_1(A)) & C = k1_domain_1(k2_zfmisc_1(u2_turing_1(A),u2_turing_1(B)),u1_turing_1(A),k1_domain_1(u2_turing_1(A),u2_turing_1(B),D,u4_turing_1(B)),E) & D != u5_turing_1(A) ) ) => k30_turing_1(A,B,C) = k22_turing_1(A,B,k8_funct_2(k2_zfmisc_1(u2_turing_1(A),u1_turing_1(A)),k3_zfmisc_1(u2_turing_1(A),u1_turing_1(A),k8_domain_1(k1_numbers,k1_real_1(1),0,1)),u3_turing_1(A),k1_domain_1(u2_turing_1(A),u1_turing_1(A),k26_turing_1(A,B,C),k28_turing_1(k21_turing_1(A,B),u1_turing_1(A),u1_turing_1(B),C)))) ) & ( ? [D] : ( m1_subset_1(D,u2_turing_1(B)) & ? [E] : ( m1_subset_1(E,u1_turing_1(B)) & C = k1_domain_1(k2_zfmisc_1(u2_turing_1(A),u2_turing_1(B)),u1_turing_1(B),k1_domain_1(u2_turing_1(A),u2_turing_1(B),u5_turing_1(A),D),E) ) ) => k30_turing_1(A,B,C) = k23_turing_1(A,B,k8_funct_2(k2_zfmisc_1(u2_turing_1(B),u1_turing_1(B)),k3_zfmisc_1(u2_turing_1(B),u1_turing_1(B),k8_domain_1(k1_numbers,k1_real_1(1),0,1)),u3_turing_1(B),k1_domain_1(u2_turing_1(B),u1_turing_1(B),k27_turing_1(A,B,C),k29_turing_1(k21_turing_1(A,B),u1_turing_1(A),u1_turing_1(B),C)))) ) & ( ( ! [D] : ( m1_subset_1(D,u2_turing_1(A)) => ! [E] : ( m1_subset_1(E,u1_turing_1(A)) => ~ ( C = k1_domain_1(k2_zfmisc_1(u2_turing_1(A),u2_turing_1(B)),u1_turing_1(A),k1_domain_1(u2_turing_1(A),u2_turing_1(B),D,u4_turing_1(B)),E) & D != u5_turing_1(A) ) ) ) & ! [D] : ( m1_subset_1(D,u2_turing_1(B)) => ! [E] : ( m1_subset_1(E,u1_turing_1(B)) => C != k1_domain_1(k2_zfmisc_1(u2_turing_1(A),u2_turing_1(B)),u1_turing_1(B),k1_domain_1(u2_turing_1(A),u2_turing_1(B),u5_turing_1(A),D),E) ) ) ) => k30_turing_1(A,B,C) = k4_domain_1(k21_turing_1(A,B),k2_xboole_0(u1_turing_1(A),u1_turing_1(B)),k1_numbers,k2_domain_1(k21_turing_1(A,B),k2_xboole_0(u1_turing_1(A),u1_turing_1(B)),C),k3_domain_1(k21_turing_1(A,B),k2_xboole_0(u1_turing_1(A),u1_turing_1(B)),C),k1_real_1(1)) ) ) ) ) ) ), file(turing_1,d30_turing_1), []). fof(d31_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => ! [B] : ( l1_turing_1(B) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k2_zfmisc_1(k21_turing_1(A,B),k2_xboole_0(u1_turing_1(A),u1_turing_1(B))),k3_zfmisc_1(k21_turing_1(A,B),k2_xboole_0(u1_turing_1(A),u1_turing_1(B)),k8_domain_1(k1_numbers,k1_real_1(1),0,1))) & m2_relset_1(C,k2_zfmisc_1(k21_turing_1(A,B),k2_xboole_0(u1_turing_1(A),u1_turing_1(B))),k3_zfmisc_1(k21_turing_1(A,B),k2_xboole_0(u1_turing_1(A),u1_turing_1(B)),k8_domain_1(k1_numbers,k1_real_1(1),0,1))) ) => ( C = k31_turing_1(A,B) <=> ! [D] : ( m1_subset_1(D,k2_zfmisc_1(k21_turing_1(A,B),k2_xboole_0(u1_turing_1(A),u1_turing_1(B)))) => k8_funct_2(k2_zfmisc_1(k21_turing_1(A,B),k2_xboole_0(u1_turing_1(A),u1_turing_1(B))),k3_zfmisc_1(k21_turing_1(A,B),k2_xboole_0(u1_turing_1(A),u1_turing_1(B)),k8_domain_1(k1_numbers,k1_real_1(1),0,1)),C,D) = k30_turing_1(A,B,D) ) ) ) ) ) ), file(turing_1,d31_turing_1), []). fof(d32_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => ! [B] : ( l1_turing_1(B) => ! [C] : ( ( v1_turing_1(C) & l1_turing_1(C) ) => ( C = k32_turing_1(A,B) <=> ( u1_turing_1(C) = k2_xboole_0(u1_turing_1(A),u1_turing_1(B)) & u2_turing_1(C) = k21_turing_1(A,B) & u3_turing_1(C) = k31_turing_1(A,B) & u4_turing_1(C) = k1_domain_1(u2_turing_1(A),u2_turing_1(B),u4_turing_1(A),u4_turing_1(B)) & u5_turing_1(C) = k1_domain_1(u2_turing_1(A),u2_turing_1(B),u5_turing_1(A),u5_turing_1(B)) ) ) ) ) ) ), file(turing_1,d32_turing_1), []). fof(d5_tarski,axiom,( ! [A,B] : k4_tarski(A,B) = k2_tarski(k2_tarski(A,B),k1_tarski(A)) ), file(tarski,d5_tarski), []). fof(dt_g1_turing_1,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(A) & v1_finset_1(A) & ~ v1_xboole_0(B) & v1_finset_1(B) & v1_funct_1(C) & v1_funct_2(C,k2_zfmisc_1(B,A),k3_zfmisc_1(B,A,k8_domain_1(k1_numbers,k1_real_1(1),0,1))) & m1_relset_1(C,k2_zfmisc_1(B,A),k3_zfmisc_1(B,A,k8_domain_1(k1_numbers,k1_real_1(1),0,1))) & m1_subset_1(D,B) & m1_subset_1(E,B) ) => ( v1_turing_1(g1_turing_1(A,B,C,D,E)) & l1_turing_1(g1_turing_1(A,B,C,D,E)) ) ) ), file(turing_1,g1_turing_1), []). fof(dt_k1_domain_1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(C,A) & m1_subset_1(D,B) ) => m1_subset_1(k1_domain_1(A,B,C,D),k2_zfmisc_1(A,B)) ) ), file(domain_1,k1_domain_1), []). fof(dt_k1_enumset1,axiom,( $true ), file(enumset1,k1_enumset1), []). fof(dt_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), []). fof(dt_k1_mcart_1,axiom,( $true ), file(mcart_1,k1_mcart_1), []). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), []). fof(dt_k1_real_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => m1_subset_1(k1_real_1(A),k1_numbers) ) ), file(real_1,k1_real_1), []). fof(dt_k1_tarski,axiom,( $true ), file(tarski,k1_tarski), []). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), []). fof(dt_k1_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), []). fof(dt_k21_turing_1,axiom,( ! [A,B] : ( ( l1_turing_1(A) & l1_turing_1(B) ) => ( ~ v1_xboole_0(k21_turing_1(A,B)) & v1_finset_1(k21_turing_1(A,B)) ) ) ), file(turing_1,k21_turing_1), []). fof(dt_k22_turing_1,axiom,( ! [A,B,C] : ( ( l1_turing_1(A) & l1_turing_1(B) & m1_subset_1(C,k3_zfmisc_1(u2_turing_1(A),u1_turing_1(A),k8_domain_1(k1_numbers,k1_real_1(1),0,1))) ) => m1_subset_1(k22_turing_1(A,B,C),k3_zfmisc_1(k21_turing_1(A,B),k2_xboole_0(u1_turing_1(A),u1_turing_1(B)),k8_domain_1(k1_numbers,k1_real_1(1),0,1))) ) ), file(turing_1,k22_turing_1), []). fof(dt_k23_turing_1,axiom,( ! [A,B,C] : ( ( l1_turing_1(A) & l1_turing_1(B) & m1_subset_1(C,k3_zfmisc_1(u2_turing_1(B),u1_turing_1(B),k8_domain_1(k1_numbers,k1_real_1(1),0,1))) ) => m1_subset_1(k23_turing_1(A,B,C),k3_zfmisc_1(k21_turing_1(A,B),k2_xboole_0(u1_turing_1(A),u1_turing_1(B)),k8_domain_1(k1_numbers,k1_real_1(1),0,1))) ) ), file(turing_1,k23_turing_1), []). fof(dt_k24_turing_1,axiom,( ! [A,B,C] : ( ( l1_turing_1(A) & l1_turing_1(B) & m1_subset_1(C,k21_turing_1(A,B)) ) => m1_subset_1(k24_turing_1(A,B,C),u2_turing_1(A)) ) ), file(turing_1,k24_turing_1), []). fof(dt_k25_turing_1,axiom,( ! [A,B,C] : ( ( l1_turing_1(A) & l1_turing_1(B) & m1_subset_1(C,k21_turing_1(A,B)) ) => m1_subset_1(k25_turing_1(A,B,C),u2_turing_1(B)) ) ), file(turing_1,k25_turing_1), []). fof(dt_k26_turing_1,axiom,( ! [A,B,C] : ( ( l1_turing_1(A) & l1_turing_1(B) & m1_subset_1(C,k2_zfmisc_1(k21_turing_1(A,B),k2_xboole_0(u1_turing_1(A),u1_turing_1(B)))) ) => m1_subset_1(k26_turing_1(A,B,C),u2_turing_1(A)) ) ), file(turing_1,k26_turing_1), []). fof(dt_k27_turing_1,axiom,( ! [A,B,C] : ( ( l1_turing_1(A) & l1_turing_1(B) & m1_subset_1(C,k2_zfmisc_1(k21_turing_1(A,B),k2_xboole_0(u1_turing_1(A),u1_turing_1(B)))) ) => m1_subset_1(k27_turing_1(A,B,C),u2_turing_1(B)) ) ), file(turing_1,k27_turing_1), []). fof(dt_k28_turing_1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & ~ v1_xboole_0(C) & m1_subset_1(D,k2_zfmisc_1(A,k2_xboole_0(B,C))) ) => m1_subset_1(k28_turing_1(A,B,C,D),B) ) ), file(turing_1,k28_turing_1), []). fof(dt_k29_turing_1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & ~ v1_xboole_0(C) & m1_subset_1(D,k2_zfmisc_1(A,k2_xboole_0(B,C))) ) => m1_subset_1(k29_turing_1(A,B,C,D),C) ) ), file(turing_1,k29_turing_1), []). fof(dt_k2_domain_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(C,k2_zfmisc_1(A,B)) ) => m1_subset_1(k2_domain_1(A,B,C),A) ) ), file(domain_1,k2_domain_1), []). fof(dt_k2_mcart_1,axiom,( $true ), file(mcart_1,k2_mcart_1), []). fof(dt_k2_tarski,axiom,( $true ), file(tarski,k2_tarski), []). fof(dt_k2_xboole_0,axiom,( $true ), file(xboole_0,k2_xboole_0), []). fof(dt_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), []). fof(dt_k30_turing_1,axiom,( ! [A,B,C] : ( ( l1_turing_1(A) & l1_turing_1(B) & m1_subset_1(C,k2_zfmisc_1(k21_turing_1(A,B),k2_xboole_0(u1_turing_1(A),u1_turing_1(B)))) ) => m1_subset_1(k30_turing_1(A,B,C),k3_zfmisc_1(k21_turing_1(A,B),k2_xboole_0(u1_turing_1(A),u1_turing_1(B)),k8_domain_1(k1_numbers,k1_real_1(1),0,1))) ) ), file(turing_1,k30_turing_1), []). fof(dt_k31_turing_1,axiom,( ! [A,B] : ( ( l1_turing_1(A) & l1_turing_1(B) ) => ( v1_funct_1(k31_turing_1(A,B)) & v1_funct_2(k31_turing_1(A,B),k2_zfmisc_1(k21_turing_1(A,B),k2_xboole_0(u1_turing_1(A),u1_turing_1(B))),k3_zfmisc_1(k21_turing_1(A,B),k2_xboole_0(u1_turing_1(A),u1_turing_1(B)),k8_domain_1(k1_numbers,k1_real_1(1),0,1))) & m2_relset_1(k31_turing_1(A,B),k2_zfmisc_1(k21_turing_1(A,B),k2_xboole_0(u1_turing_1(A),u1_turing_1(B))),k3_zfmisc_1(k21_turing_1(A,B),k2_xboole_0(u1_turing_1(A),u1_turing_1(B)),k8_domain_1(k1_numbers,k1_real_1(1),0,1))) ) ) ), file(turing_1,k31_turing_1), []). fof(dt_k32_turing_1,axiom,( ! [A,B] : ( ( l1_turing_1(A) & l1_turing_1(B) ) => ( v1_turing_1(k32_turing_1(A,B)) & l1_turing_1(k32_turing_1(A,B)) ) ) ), file(turing_1,k32_turing_1), []). fof(dt_k3_domain_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(C,k2_zfmisc_1(A,B)) ) => m1_subset_1(k3_domain_1(A,B,C),B) ) ), file(domain_1,k3_domain_1), []). fof(dt_k3_mcart_1,axiom,( $true ), file(mcart_1,k3_mcart_1), []). fof(dt_k3_zfmisc_1,axiom,( $true ), file(zfmisc_1,k3_zfmisc_1), []). fof(dt_k4_domain_1,axiom,( ! [A,B,C,D,E,F] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & ~ v1_xboole_0(C) & m1_subset_1(D,A) & m1_subset_1(E,B) & m1_subset_1(F,C) ) => m1_subset_1(k4_domain_1(A,B,C,D,E,F),k3_zfmisc_1(A,B,C)) ) ), file(domain_1,k4_domain_1), []). fof(dt_k4_tarski,axiom,( $true ), file(tarski,k4_tarski), []). fof(dt_k4_xcmplx_0,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_xcmplx_0(k4_xcmplx_0(A)) ) ), file(xcmplx_0,k4_xcmplx_0), []). fof(dt_k5_mcart_1,axiom,( ! [A,B,C,D] : ( m1_subset_1(D,k3_zfmisc_1(A,B,C)) => m1_subset_1(k5_mcart_1(A,B,C,D),A) ) ), file(mcart_1,k5_mcart_1), []). fof(dt_k5_numbers,axiom,( m1_subset_1(k5_numbers,k1_zfmisc_1(k1_numbers)) ), file(numbers,k5_numbers), []). fof(dt_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), []). fof(dt_k6_domain_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => m1_subset_1(k6_domain_1(A,B),k1_zfmisc_1(A)) ) ), file(domain_1,k6_domain_1), []). fof(dt_k6_mcart_1,axiom,( ! [A,B,C,D] : ( m1_subset_1(D,k3_zfmisc_1(A,B,C)) => m1_subset_1(k6_mcart_1(A,B,C,D),B) ) ), file(mcart_1,k6_mcart_1), []). fof(dt_k7_mcart_1,axiom,( ! [A,B,C,D] : ( m1_subset_1(D,k3_zfmisc_1(A,B,C)) => m1_subset_1(k7_mcart_1(A,B,C,D),C) ) ), file(mcart_1,k7_mcart_1), []). fof(dt_k8_domain_1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) & m1_subset_1(C,A) & m1_subset_1(D,A) ) => m1_subset_1(k8_domain_1(A,B,C,D),k1_zfmisc_1(A)) ) ), file(domain_1,k8_domain_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_turing_1,axiom,( $true ), file(turing_1,l1_turing_1), []). fof(dt_m1_relset_1,axiom,( $true ), file(relset_1,m1_relset_1), []). fof(dt_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), []). fof(dt_m2_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_u1_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => ( ~ v1_xboole_0(u1_turing_1(A)) & v1_finset_1(u1_turing_1(A)) ) ) ), file(turing_1,u1_turing_1), []). fof(dt_u2_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => ( ~ v1_xboole_0(u2_turing_1(A)) & v1_finset_1(u2_turing_1(A)) ) ) ), file(turing_1,u2_turing_1), []). fof(dt_u3_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => ( v1_funct_1(u3_turing_1(A)) & v1_funct_2(u3_turing_1(A),k2_zfmisc_1(u2_turing_1(A),u1_turing_1(A)),k3_zfmisc_1(u2_turing_1(A),u1_turing_1(A),k8_domain_1(k1_numbers,k1_real_1(1),0,1))) & m2_relset_1(u3_turing_1(A),k2_zfmisc_1(u2_turing_1(A),u1_turing_1(A)),k3_zfmisc_1(u2_turing_1(A),u1_turing_1(A),k8_domain_1(k1_numbers,k1_real_1(1),0,1))) ) ) ), file(turing_1,u3_turing_1), []). fof(dt_u4_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => m1_subset_1(u4_turing_1(A),u2_turing_1(A)) ) ), file(turing_1,u4_turing_1), []). fof(dt_u5_turing_1,axiom,( ! [A] : ( l1_turing_1(A) => m1_subset_1(u5_turing_1(A),u2_turing_1(A)) ) ), file(turing_1,u5_turing_1), []). fof(existence_l1_turing_1,axiom,( ? [A] : l1_turing_1(A) ), file(turing_1,l1_turing_1), []). fof(existence_m1_relset_1,axiom,( ! [A,B] : ? [C] : m1_relset_1(C,A,B) ), file(relset_1,m1_relset_1), []). fof(existence_m1_subset_1,axiom,( ! [A] : ? [B] : m1_subset_1(B,A) ), file(subset_1,m1_subset_1), []). fof(existence_m2_relset_1,axiom,( ! [A,B] : ? [C] : m2_relset_1(C,A,B) ), file(relset_1,m2_relset_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(fc15_finset_1,axiom,( ! [A,B,C] : ( ( v1_finset_1(A) & v1_finset_1(B) & v1_finset_1(C) ) => v1_finset_1(k3_zfmisc_1(A,B,C)) ) ), file(finset_1,fc15_finset_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_numbers,axiom,( ~ v1_xboole_0(k1_numbers) ), file(numbers,fc1_numbers), []). 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_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ), file(subset_1,fc1_subset_1), []). fof(fc1_xboole_0,axiom,( v1_xboole_0(k1_xboole_0) ), file(xboole_0,fc1_xboole_0), []). fof(fc2_comput_1,axiom, ( v1_xboole_0(k1_xboole_0) & v1_relat_1(k1_xboole_0) & v3_relat_1(k1_xboole_0) & v1_funct_1(k1_xboole_0) & v2_funct_1(k1_xboole_0) & v1_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_funct_7(k1_xboole_0) & v4_comput_1(k1_xboole_0) ), file(comput_1,fc2_comput_1), []). fof(fc2_finset_1,axiom,( ! [A,B] : ( ~ v1_xboole_0(k2_tarski(A,B)) & v1_finset_1(k2_tarski(A,B)) ) ), file(finset_1,fc2_finset_1), []). fof(fc2_mod_2,axiom,( ! [A,B,C] : ~ v1_xboole_0(k1_enumset1(A,B,C)) ), file(mod_2,fc2_mod_2), []). fof(fc2_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_tarski(A)) ), file(subset_1,fc2_subset_1), []). fof(fc2_xboole_0,axiom,( ! [A,B] : ( ~ v1_xboole_0(A) => ~ v1_xboole_0(k2_xboole_0(A,B)) ) ), file(xboole_0,fc2_xboole_0), []). fof(fc3_finset_1,axiom,( ! [A,B,C] : v1_finset_1(k1_enumset1(A,B,C)) ), file(finset_1,fc3_finset_1), []). fof(fc3_int_1,axiom,( ! [A] : ( v1_int_1(A) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & v1_int_1(k4_xcmplx_0(A)) ) ) ), file(int_1,fc3_int_1), []). fof(fc3_subset_1,axiom,( ! [A,B] : ~ v1_xboole_0(k2_tarski(A,B)) ), file(subset_1,fc3_subset_1), []). fof(fc3_xboole_0,axiom,( ! [A,B] : ( ~ v1_xboole_0(A) => ~ v1_xboole_0(k2_xboole_0(B,A)) ) ), file(xboole_0,fc3_xboole_0), []). fof(fc4_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ), file(subset_1,fc4_subset_1), []). fof(fc5_comput_1,axiom, ( v1_xboole_0(k1_xboole_0) & v1_relat_1(k1_xboole_0) & v3_relat_1(k1_xboole_0) & v1_funct_1(k1_xboole_0) & v2_funct_1(k1_xboole_0) & v1_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_funct_7(k1_xboole_0) & v4_comput_1(k1_xboole_0) & v5_comput_1(k1_xboole_0) ), file(comput_1,fc5_comput_1), []). fof(fc5_int_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & ~ v2_xreal_0(k4_xcmplx_0(A)) & v1_int_1(k4_xcmplx_0(A)) ) ) ), file(int_1,fc5_int_1), []). fof(fc5_subset_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & ~ v1_xboole_0(C) ) => ~ v1_xboole_0(k3_zfmisc_1(A,B,C)) ) ), file(subset_1,fc5_subset_1), []). 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(free_g1_turing_1,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(A) & v1_finset_1(A) & ~ v1_xboole_0(B) & v1_finset_1(B) & v1_funct_1(C) & v1_funct_2(C,k2_zfmisc_1(B,A),k3_zfmisc_1(B,A,k8_domain_1(k1_numbers,k1_real_1(1),0,1))) & m1_relset_1(C,k2_zfmisc_1(B,A),k3_zfmisc_1(B,A,k8_domain_1(k1_numbers,k1_real_1(1),0,1))) & m1_subset_1(D,B) & m1_subset_1(E,B) ) => ! [F,G,H,I,J] : ( g1_turing_1(A,B,C,D,E) = g1_turing_1(F,G,H,I,J) => ( A = F & B = G & C = H & D = I & E = J ) ) ) ), file(turing_1,g1_turing_1), []). fof(idempotence_k2_xboole_0,axiom,( ! [A,B] : k2_xboole_0(A,A) = A ), file(xboole_0,k2_xboole_0), []). fof(involutiveness_k1_real_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => k1_real_1(k1_real_1(A)) = A ) ), file(real_1,k1_real_1), []). fof(involutiveness_k4_xcmplx_0,axiom,( ! [A] : ( v1_xcmplx_0(A) => k4_xcmplx_0(k4_xcmplx_0(A)) = A ) ), file(xcmplx_0,k4_xcmplx_0), []). fof(rc1_arytm_3,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ), file(arytm_3,rc1_arytm_3), []). fof(rc1_finset_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) ) ), file(finset_1,rc1_finset_1), []). fof(rc1_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc1_funct_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_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_turing_1,axiom,( ? [A] : ( l1_turing_1(A) & v1_turing_1(A) ) ), file(turing_1,rc1_turing_1), []). fof(rc1_xboole_0,axiom,( ? [A] : v1_xboole_0(A) ), file(xboole_0,rc1_xboole_0), []). fof(rc2_finset_1,axiom,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v2_funct_1(B) & v1_ordinal1(B) & v2_ordinal1(B) & v3_ordinal1(B) & v4_ordinal2(B) & v1_finset_1(B) ) ), file(finset_1,rc2_finset_1), []). fof(rc2_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) ), file(funct_1,rc2_funct_1), []). fof(rc2_int_1,axiom,( ? [A] : v1_int_1(A) ), file(int_1,rc2_int_1), []). fof(rc2_subset_1,axiom,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) ) ), file(subset_1,rc2_subset_1), []). fof(rc2_xboole_0,axiom,( ? [A] : ~ v1_xboole_0(A) ), file(xboole_0,rc2_xboole_0), []). fof(rc3_finset_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & v1_finset_1(B) ) ) ), file(finset_1,rc3_finset_1), []). fof(rc3_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ), file(funct_1,rc3_funct_1), []). fof(rc4_comput_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v4_comput_1(A) ) ), file(comput_1,rc4_comput_1), []). fof(rc4_finset_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & v1_finset_1(B) ) ) ), file(finset_1,rc4_finset_1), []). fof(rc4_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v3_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc4_funct_1), []). fof(rc9_comput_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) & v5_comput_1(A) ) ), file(comput_1,rc9_comput_1), []). fof(redefinition_k1_domain_1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(C,A) & m1_subset_1(D,B) ) => k1_domain_1(A,B,C,D) = k4_tarski(C,D) ) ), file(domain_1,k1_domain_1), []). fof(redefinition_k1_real_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => k1_real_1(A) = k4_xcmplx_0(A) ) ), file(real_1,k1_real_1), []). fof(redefinition_k24_turing_1,axiom,( ! [A,B,C] : ( ( l1_turing_1(A) & l1_turing_1(B) & m1_subset_1(C,k21_turing_1(A,B)) ) => k24_turing_1(A,B,C) = k1_mcart_1(C) ) ), file(turing_1,k24_turing_1), []). fof(redefinition_k25_turing_1,axiom,( ! [A,B,C] : ( ( l1_turing_1(A) & l1_turing_1(B) & m1_subset_1(C,k21_turing_1(A,B)) ) => k25_turing_1(A,B,C) = k2_mcart_1(C) ) ), file(turing_1,k25_turing_1), []). fof(redefinition_k2_domain_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(C,k2_zfmisc_1(A,B)) ) => k2_domain_1(A,B,C) = k1_mcart_1(C) ) ), file(domain_1,k2_domain_1), []). fof(redefinition_k3_domain_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(C,k2_zfmisc_1(A,B)) ) => k3_domain_1(A,B,C) = k2_mcart_1(C) ) ), file(domain_1,k3_domain_1), []). fof(redefinition_k4_domain_1,axiom,( ! [A,B,C,D,E,F] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & ~ v1_xboole_0(C) & m1_subset_1(D,A) & m1_subset_1(E,B) & m1_subset_1(F,C) ) => k4_domain_1(A,B,C,D,E,F) = k3_mcart_1(D,E,F) ) ), file(domain_1,k4_domain_1), []). fof(redefinition_k5_numbers,axiom,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), []). fof(redefinition_k6_domain_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => k6_domain_1(A,B) = k1_tarski(B) ) ), file(domain_1,k6_domain_1), []). fof(redefinition_k8_domain_1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) & m1_subset_1(C,A) & m1_subset_1(D,A) ) => k8_domain_1(A,B,C,D) = k1_enumset1(B,C,D) ) ), file(domain_1,k8_domain_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_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(reflexivity_r1_tarski,axiom,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), []). fof(spc0_boole,axiom,( v1_xboole_0(0) ), file(boole,spc0_boole), []). fof(spc1_boole,axiom,( ~ v1_xboole_0(1) ), file(boole,spc1_boole), []). fof(spc1_numerals,axiom, ( v2_xreal_0(1) & m1_subset_1(1,k5_numbers) ), file(numerals,spc1_numerals), []). fof(t1_boole,axiom,( ! [A] : k2_xboole_0(A,k1_xboole_0) = A ), file(boole,t1_boole), []). fof(t1_numerals,axiom,( m1_subset_1(0,k5_numbers) ), file(numerals,t1_numerals), []). fof(t1_subset,axiom,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), []). fof(t2_subset,axiom,( ! [A,B] : ( m1_subset_1(A,B) => ( v1_xboole_0(B) | r2_hidden(A,B) ) ) ), file(subset,t2_subset), []). fof(t3_subset,axiom,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), []). 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_subset,axiom,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), []). fof(t6_boole,axiom,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), []). fof(t7_boole,axiom,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), []). fof(t8_boole,axiom,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), []).