% Mizar ND problem: t23_ami_5,ami_5,103,63 fof(rc3_funct_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ), file(funct_1,rc3_funct_1), [interesting(0.9),axiom,file(funct_1,rc3_funct_1)]). fof(cc1_fraenkel,theorem,( ! [A] : ( v1_fraenkel(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_relat_1(B) & v1_funct_1(B) ) ) ) ), file(fraenkel,cc1_fraenkel), [interesting(0.9),axiom,file(fraenkel,cc1_fraenkel)]). fof(cc2_funct_1,theorem,( ! [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), [interesting(0.9),axiom,file(funct_1,cc2_funct_1)]). fof(fc3_ami_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => v1_fraenkel(k4_card_3(A)) ) ), file(ami_1,fc3_ami_1), [interesting(0.9),axiom,file(ami_1,fc3_ami_1)]). fof(rc1_fraenkel,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_fraenkel(A) ) ), file(fraenkel,rc1_fraenkel), [interesting(0.9),axiom,file(fraenkel,rc1_fraenkel)]). fof(rc1_funct_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc1_funct_1), [interesting(0.9),axiom,file(funct_1,rc1_funct_1)]). fof(rc2_funct_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) ), file(funct_1,rc2_funct_1), [interesting(0.9),axiom,file(funct_1,rc2_funct_1)]). fof(commutativity_k2_tarski,theorem,( ! [A,B] : k2_tarski(A,B) = k2_tarski(B,A) ), file(tarski,k2_tarski), [interesting(0.9),axiom,file(tarski,k2_tarski)]). fof(redefinition_m2_relset_1,definition,( ! [A,B,C] : ( m2_relset_1(C,A,B) <=> m1_relset_1(C,A,B) ) ), file(relset_1,m2_relset_1), [interesting(0.9),axiom,file(relset_1,m2_relset_1)]). fof(dt_k13_finseq_1,axiom,( $true ), file(finseq_1,k13_finseq_1), [interesting(0.9),axiom,file(finseq_1,k13_finseq_1)]). fof(dt_k1_funct_2,axiom,( $true ), file(funct_2,k1_funct_2), [interesting(0.9),axiom,file(funct_2,k1_funct_2)]). fof(dt_k2_tarski,axiom,( $true ), file(tarski,k2_tarski), [interesting(0.9),axiom,file(tarski,k2_tarski)]). fof(dt_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,k2_zfmisc_1)]). fof(dt_k3_tarski,axiom,( $true ), file(tarski,k3_tarski), [interesting(0.9),axiom,file(tarski,k3_tarski)]). fof(dt_k4_card_3,axiom,( $true ), file(card_3,k4_card_3), [interesting(0.9),axiom,file(card_3,k4_card_3)]). fof(dt_m1_relset_1,axiom,( $true ), file(relset_1,m1_relset_1), [interesting(0.9),axiom,file(relset_1,m1_relset_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), [interesting(0.9),axiom,file(relset_1,m2_relset_1)]). fof(cc1_relset_1,theorem,( ! [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), [interesting(0.9),axiom,file(relset_1,cc1_relset_1)]). fof(fc16_finseq_1,theorem,( ! [A] : ( ~ v1_xboole_0(k13_finseq_1(A)) & v1_fraenkel(k13_finseq_1(A)) ) ), file(finseq_1,fc16_finseq_1), [interesting(0.9),axiom,file(finseq_1,fc16_finseq_1)]). fof(fc1_fraenkel,theorem,( ! [A,B] : v1_fraenkel(k1_funct_2(A,B)) ), file(fraenkel,fc1_fraenkel), [interesting(0.9),axiom,file(fraenkel,fc1_fraenkel)]). fof(fc3_setfam_1,theorem,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ( ~ v1_xboole_0(k2_tarski(A,B)) & v1_setfam_1(k2_tarski(A,B)) ) ) ), file(setfam_1,fc3_setfam_1), [interesting(0.9),axiom,file(setfam_1,fc3_setfam_1)]). fof(fc4_ami_1,theorem,( ! [A,B,C] : ( ( v1_setfam_1(B) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) ) => ( ~ v1_xboole_0(k4_card_3(C)) & v1_fraenkel(k4_card_3(C)) ) ) ), file(ami_1,fc4_ami_1), [interesting(0.9),axiom,file(ami_1,fc4_ami_1)]). fof(fc9_finseq_1,theorem,( ! [A] : ~ v1_xboole_0(k13_finseq_1(A)) ), file(finseq_1,fc9_finseq_1), [interesting(0.9),axiom,file(finseq_1,fc9_finseq_1)]). fof(free_g1_ami_1,definition,( ! [A,B,C,D,E,F,G,H] : ( ( m1_subset_1(C,B) & m1_subset_1(D,k1_zfmisc_1(B)) & ~ v1_xboole_0(E) & ~ v1_xboole_0(F) & m1_subset_1(F,k1_zfmisc_1(k2_zfmisc_1(E,k13_finseq_1(k2_xboole_0(k3_tarski(A),B))))) & v1_funct_1(G) & v1_funct_2(G,B,k2_xboole_0(A,k2_tarski(F,D))) & m1_relset_1(G,B,k2_xboole_0(A,k2_tarski(F,D))) & v1_funct_1(H) & v1_funct_2(H,F,k1_funct_2(k4_card_3(G),k4_card_3(G))) & m1_relset_1(H,F,k1_funct_2(k4_card_3(G),k4_card_3(G))) ) => ! [I,J,K,L,M,N,O,P] : ( g1_ami_1(A,B,C,D,E,F,G,H) = g1_ami_1(I,J,K,L,M,N,O,P) => ( A = I & B = J & C = K & D = L & E = M & F = N & G = O & H = P ) ) ) ), file(ami_1,g1_ami_1), [interesting(0.9),axiom,file(ami_1,g1_ami_1)]). fof(dt_g1_ami_1,axiom,( ! [A,B,C,D,E,F,G,H] : ( ( m1_subset_1(C,B) & m1_subset_1(D,k1_zfmisc_1(B)) & ~ v1_xboole_0(E) & ~ v1_xboole_0(F) & m1_subset_1(F,k1_zfmisc_1(k2_zfmisc_1(E,k13_finseq_1(k2_xboole_0(k3_tarski(A),B))))) & v1_funct_1(G) & v1_funct_2(G,B,k2_xboole_0(A,k2_tarski(F,D))) & m1_relset_1(G,B,k2_xboole_0(A,k2_tarski(F,D))) & v1_funct_1(H) & v1_funct_2(H,F,k1_funct_2(k4_card_3(G),k4_card_3(G))) & m1_relset_1(H,F,k1_funct_2(k4_card_3(G),k4_card_3(G))) ) => ( v1_ami_1(g1_ami_1(A,B,C,D,E,F,G,H),A) & l1_ami_1(g1_ami_1(A,B,C,D,E,F,G,H),A) ) ) ), file(ami_1,g1_ami_1), [interesting(0.9),axiom,file(ami_1,g1_ami_1)]). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), [interesting(0.9),axiom,file(numbers,k1_numbers)]). fof(dt_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), [interesting(0.9),axiom,file(ordinal2,k5_ordinal2)]). fof(dt_u1_ami_1,axiom,( ! [A,B] : ( l1_ami_1(B,A) => m1_subset_1(u1_ami_1(A,B),u1_struct_0(B)) ) ), file(ami_1,u1_ami_1), [interesting(0.9),axiom,file(ami_1,u1_ami_1)]). fof(dt_u2_ami_1,axiom,( ! [A,B] : ( l1_ami_1(B,A) => m1_subset_1(u2_ami_1(A,B),k1_zfmisc_1(u1_struct_0(B))) ) ), file(ami_1,u2_ami_1), [interesting(0.9),axiom,file(ami_1,u2_ami_1)]). fof(dt_u3_ami_1,axiom,( ! [A,B] : ( l1_ami_1(B,A) => ~ v1_xboole_0(u3_ami_1(A,B)) ) ), file(ami_1,u3_ami_1), [interesting(0.9),axiom,file(ami_1,u3_ami_1)]). fof(dt_u4_ami_1,axiom,( ! [A,B] : ( l1_ami_1(B,A) => ( ~ v1_xboole_0(u4_ami_1(A,B)) & m1_subset_1(u4_ami_1(A,B),k1_zfmisc_1(k2_zfmisc_1(u3_ami_1(A,B),k13_finseq_1(k2_xboole_0(k3_tarski(A),u1_struct_0(B)))))) ) ) ), file(ami_1,u4_ami_1), [interesting(0.9),axiom,file(ami_1,u4_ami_1)]). fof(dt_u5_ami_1,axiom,( ! [A,B] : ( l1_ami_1(B,A) => ( v1_funct_1(u5_ami_1(A,B)) & v1_funct_2(u5_ami_1(A,B),u1_struct_0(B),k2_xboole_0(A,k2_tarski(u4_ami_1(A,B),u2_ami_1(A,B)))) & m2_relset_1(u5_ami_1(A,B),u1_struct_0(B),k2_xboole_0(A,k2_tarski(u4_ami_1(A,B),u2_ami_1(A,B)))) ) ) ), file(ami_1,u5_ami_1), [interesting(0.9),axiom,file(ami_1,u5_ami_1)]). fof(dt_u6_ami_1,axiom,( ! [A,B] : ( l1_ami_1(B,A) => ( v1_funct_1(u6_ami_1(A,B)) & v1_funct_2(u6_ami_1(A,B),u4_ami_1(A,B),k1_funct_2(k4_card_3(u5_ami_1(A,B)),k4_card_3(u5_ami_1(A,B)))) & m2_relset_1(u6_ami_1(A,B),u4_ami_1(A,B),k1_funct_2(k4_card_3(u5_ami_1(A,B)),k4_card_3(u5_ami_1(A,B)))) ) ) ), file(ami_1,u6_ami_1), [interesting(0.9),axiom,file(ami_1,u6_ami_1)]). fof(cc1_arytm_3,theorem,( ! [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), [interesting(0.9),axiom,file(arytm_3,cc1_arytm_3)]). fof(cc1_setfam_1,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_setfam_1(A) ) => ! [B] : ( m1_subset_1(B,A) => ~ v1_xboole_0(B) ) ) ), file(setfam_1,cc1_setfam_1), [interesting(0.9),axiom,file(setfam_1,cc1_setfam_1)]). fof(cc2_arytm_3,theorem,( ! [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), [interesting(0.9),axiom,file(arytm_3,cc2_arytm_3)]). fof(cc3_arytm_3,theorem,( ! [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), [interesting(0.9),axiom,file(arytm_3,cc3_arytm_3)]). fof(cc3_int_1,theorem,( ! [A] : ( v4_ordinal2(A) => v1_int_1(A) ) ), file(int_1,cc3_int_1), [interesting(0.9),axiom,file(int_1,cc3_int_1)]). fof(cc4_int_1,theorem,( ! [A] : ( v1_int_1(A) => ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(int_1,cc4_int_1), [interesting(0.9),axiom,file(int_1,cc4_int_1)]). fof(fc1_numbers,theorem,( ~ v1_xboole_0(k1_numbers) ), file(numbers,fc1_numbers), [interesting(0.9),axiom,file(numbers,fc1_numbers)]). fof(fc1_ordinal2,theorem, ( v1_ordinal1(k5_ordinal2) & v2_ordinal1(k5_ordinal2) & v3_ordinal1(k5_ordinal2) & ~ v1_xboole_0(k5_ordinal2) ), file(ordinal2,fc1_ordinal2), [interesting(0.9),axiom,file(ordinal2,fc1_ordinal2)]). fof(fc2_ami_1,theorem,( ! [A,B] : ( ( ~ v2_ami_1(B,A) & l1_ami_1(B,A) ) => ~ v1_xboole_0(u2_ami_1(A,B)) ) ), file(ami_1,fc2_ami_1), [interesting(0.9),axiom,file(ami_1,fc2_ami_1)]). fof(fc4_setfam_1,theorem,( ! [A,B] : ( ( v1_setfam_1(A) & v1_setfam_1(B) ) => v1_setfam_1(k2_xboole_0(A,B)) ) ), file(setfam_1,fc4_setfam_1), [interesting(0.9),axiom,file(setfam_1,fc4_setfam_1)]). fof(rc1_arytm_3,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ), file(arytm_3,rc1_arytm_3), [interesting(0.9),axiom,file(arytm_3,rc1_arytm_3)]). fof(rc1_int_1,theorem,( ? [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), [interesting(0.9),axiom,file(int_1,rc1_int_1)]). fof(rc1_setfam_1,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_setfam_1(A) ) ), file(setfam_1,rc1_setfam_1), [interesting(0.9),axiom,file(setfam_1,rc1_setfam_1)]). fof(rc2_int_1,theorem,( ? [A] : v1_int_1(A) ), file(int_1,rc2_int_1), [interesting(0.9),axiom,file(int_1,rc2_int_1)]). fof(rc3_ami_1,theorem,( ! [A] : ( v1_setfam_1(A) => ? [B] : ( l1_ami_1(B,A) & ~ v2_ami_1(B,A) & v4_ami_1(B,A) ) ) ), file(ami_1,rc3_ami_1), [interesting(0.9),axiom,file(ami_1,rc3_ami_1)]). fof(rc7_ami_1,theorem,( ! [A] : ( v1_setfam_1(A) => ? [B] : ( l1_ami_1(B,A) & ~ v3_struct_0(B) & v1_ami_1(B,A) & ~ v2_ami_1(B,A) & v4_ami_1(B,A) & v5_ami_1(B,A) & v6_ami_1(B,A) & v7_ami_1(B,A) & v8_ami_1(B,A) ) ) ), file(ami_1,rc7_ami_1), [interesting(0.9),axiom,file(ami_1,rc7_ami_1)]). fof(abstractness_v1_ami_1,theorem,( ! [A,B] : ( l1_ami_1(B,A) => ( v1_ami_1(B,A) => B = g1_ami_1(A,u1_struct_0(B),u1_ami_1(A,B),u2_ami_1(A,B),u3_ami_1(A,B),u4_ami_1(A,B),u5_ami_1(A,B),u6_ami_1(A,B)) ) ) ), file(ami_1,v1_ami_1), [interesting(0.9),axiom,file(ami_1,v1_ami_1)]). fof(redefinition_k5_numbers,definition,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), [interesting(0.9),axiom,file(numbers,k5_numbers)]). fof(dt_k1_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,k1_zfmisc_1)]). fof(dt_k5_numbers,axiom,( m1_subset_1(k5_numbers,k1_zfmisc_1(k1_numbers)) ), file(numbers,k5_numbers), [interesting(0.9),axiom,file(numbers,k5_numbers)]). fof(dt_l1_ami_1,axiom,( ! [A,B] : ( l1_ami_1(B,A) => l1_struct_0(B) ) ), file(ami_1,l1_ami_1), [interesting(0.9),axiom,file(ami_1,l1_ami_1)]). fof(dt_l1_struct_0,axiom,( $true ), file(struct_0,l1_struct_0), [interesting(0.9),axiom,file(struct_0,l1_struct_0)]). fof(dt_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), [interesting(0.9),axiom,file(subset_1,m1_subset_1)]). fof(cc1_funct_1,theorem,( ! [A] : ( v1_xboole_0(A) => v1_funct_1(A) ) ), file(funct_1,cc1_funct_1), [interesting(0.9),axiom,file(funct_1,cc1_funct_1)]). fof(cc1_int_1,theorem,( ! [A] : ( m1_subset_1(A,k4_numbers) => v1_int_1(A) ) ), file(int_1,cc1_int_1), [interesting(0.9),axiom,file(int_1,cc1_int_1)]). fof(cc2_int_1,theorem,( ! [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), [interesting(0.9),axiom,file(int_1,cc2_int_1)]). fof(fc1_struct_0,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => ~ v1_xboole_0(u1_struct_0(A)) ) ), file(struct_0,fc1_struct_0), [interesting(0.9),axiom,file(struct_0,fc1_struct_0)]). fof(fc2_setfam_1,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ( ~ v1_xboole_0(k1_tarski(A)) & v1_setfam_1(k1_tarski(A)) ) ) ), file(setfam_1,fc2_setfam_1), [interesting(0.9),axiom,file(setfam_1,fc2_setfam_1)]). fof(fc2_xboole_0,theorem,( ! [A,B] : ( ~ v1_xboole_0(A) => ~ v1_xboole_0(k2_xboole_0(A,B)) ) ), file(xboole_0,fc2_xboole_0), [interesting(0.9),axiom,file(xboole_0,fc2_xboole_0)]). fof(fc3_xboole_0,theorem,( ! [A,B] : ( ~ v1_xboole_0(A) => ~ v1_xboole_0(k2_xboole_0(B,A)) ) ), file(xboole_0,fc3_xboole_0), [interesting(0.9),axiom,file(xboole_0,fc3_xboole_0)]). fof(rc1_ami_1,theorem,( ! [A] : ? [B] : ( l1_ami_1(B,A) & v1_ami_1(B,A) ) ), file(ami_1,rc1_ami_1), [interesting(0.9),axiom,file(ami_1,rc1_ami_1)]). fof(rc1_xboole_0,theorem,( ? [A] : v1_xboole_0(A) ), file(xboole_0,rc1_xboole_0), [interesting(0.9),axiom,file(xboole_0,rc1_xboole_0)]). fof(rc2_ami_1,theorem,( ! [A] : ? [B] : ( l1_ami_1(B,A) & ~ v3_struct_0(B) & ~ v2_ami_1(B,A) ) ), file(ami_1,rc2_ami_1), [interesting(0.9),axiom,file(ami_1,rc2_ami_1)]). fof(rc2_xboole_0,theorem,( ? [A] : ~ v1_xboole_0(A) ), file(xboole_0,rc2_xboole_0), [interesting(0.9),axiom,file(xboole_0,rc2_xboole_0)]). fof(rc3_struct_0,theorem,( ? [A] : ( l1_struct_0(A) & ~ v3_struct_0(A) ) ), file(struct_0,rc3_struct_0), [interesting(0.9),axiom,file(struct_0,rc3_struct_0)]). fof(rc5_ami_1,theorem,( ! [A] : ? [B] : ( l1_ami_1(B,A) & v1_ami_1(B,A) & v6_ami_1(B,A) ) ), file(ami_1,rc5_ami_1), [interesting(0.9),axiom,file(ami_1,rc5_ami_1)]). fof(rc5_struct_0,theorem,( ! [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), [interesting(0.9),axiom,file(struct_0,rc5_struct_0)]). fof(rc6_ami_1,theorem,( ! [A] : ? [B] : ( l1_ami_1(B,A) & ~ v3_struct_0(B) & v1_ami_1(B,A) & ~ v2_ami_1(B,A) & v5_ami_1(B,A) & v6_ami_1(B,A) & v8_ami_1(B,A) ) ), file(ami_1,rc6_ami_1), [interesting(0.9),axiom,file(ami_1,rc6_ami_1)]). fof(rc8_ami_1,theorem,( ! [A] : ? [B] : ( l1_ami_1(B,A) & v1_ami_1(B,A) & v6_ami_1(B,A) & v10_ami_1(B,A) ) ), file(ami_1,rc8_ami_1), [interesting(0.9),axiom,file(ami_1,rc8_ami_1)]). fof(rc9_ami_1,theorem,( ! [A] : ? [B] : ( l1_ami_1(B,A) & ~ v3_struct_0(B) & v1_ami_1(B,A) & ~ v2_ami_1(B,A) & v5_ami_1(B,A) & v6_ami_1(B,A) & v8_ami_1(B,A) & v10_ami_1(B,A) ) ), file(ami_1,rc9_ami_1), [interesting(0.9),axiom,file(ami_1,rc9_ami_1)]). fof(commutativity_k2_xboole_0,theorem,( ! [A,B] : k2_xboole_0(A,B) = k2_xboole_0(B,A) ), file(xboole_0,k2_xboole_0), [interesting(0.9),axiom,file(xboole_0,k2_xboole_0)]). fof(idempotence_k2_xboole_0,theorem,( ! [A,B] : k2_xboole_0(A,A) = A ), file(xboole_0,k2_xboole_0), [interesting(0.9),axiom,file(xboole_0,k2_xboole_0)]). fof(redefinition_k1_struct_0,definition,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & m1_subset_1(B,u1_struct_0(A)) ) => k1_struct_0(A,B) = k1_tarski(B) ) ), file(struct_0,k1_struct_0), [interesting(0.9),axiom,file(struct_0,k1_struct_0)]). fof(dt_k1_ami_3,axiom, ( v1_ami_1(k1_ami_3,k1_tarski(k4_numbers)) & l1_ami_1(k1_ami_3,k1_tarski(k4_numbers)) ), file(ami_3,k1_ami_3), [interesting(0.9),axiom,file(ami_3,k1_ami_3)]). fof(dt_k1_struct_0,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & m1_subset_1(B,u1_struct_0(A)) ) => m1_subset_1(k1_struct_0(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ), file(struct_0,k1_struct_0), [interesting(0.9),axiom,file(struct_0,k1_struct_0)]). fof(dt_k1_tarski,axiom,( $true ), file(tarski,k1_tarski), [interesting(0.9),axiom,file(tarski,k1_tarski)]). fof(dt_k2_ami_1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(B) & l1_ami_1(B,A) ) => m1_subset_1(k2_ami_1(A,B),u1_struct_0(B)) ) ), file(ami_1,k2_ami_1), [interesting(0.9),axiom,file(ami_1,k2_ami_1)]). fof(dt_k2_ami_2,axiom,( m1_subset_1(k2_ami_2,k1_zfmisc_1(k5_numbers)) ), file(ami_2,k2_ami_2), [interesting(0.9),axiom,file(ami_2,k2_ami_2)]). fof(dt_k2_xboole_0,axiom,( $true ), file(xboole_0,k2_xboole_0), [interesting(0.9),axiom,file(xboole_0,k2_xboole_0)]). fof(dt_k3_ami_2,axiom,( m1_subset_1(k3_ami_2,k1_zfmisc_1(k5_numbers)) ), file(ami_2,k3_ami_2), [interesting(0.9),axiom,file(ami_2,k3_ami_2)]). fof(dt_k4_numbers,axiom,( $true ), file(numbers,k4_numbers), [interesting(0.9),axiom,file(numbers,k4_numbers)]). fof(dt_u1_struct_0,axiom,( $true ), file(struct_0,u1_struct_0), [interesting(0.9),axiom,file(struct_0,u1_struct_0)]). fof(fc1_ami_2,theorem, ( ~ v1_xboole_0(k2_ami_2) & v1_membered(k2_ami_2) & v2_membered(k2_ami_2) & v3_membered(k2_ami_2) & v4_membered(k2_ami_2) & v5_membered(k2_ami_2) ), file(ami_2,fc1_ami_2), [interesting(0.9),axiom,file(ami_2,fc1_ami_2)]). fof(fc1_ami_3,theorem, ( ~ v3_struct_0(k1_ami_3) & v1_ami_1(k1_ami_3,k1_tarski(k4_numbers)) & ~ v2_ami_1(k1_ami_3,k1_tarski(k4_numbers)) ), file(ami_3,fc1_ami_3), [interesting(0.9),axiom,file(ami_3,fc1_ami_3)]). fof(fc2_ami_2,theorem, ( ~ v1_xboole_0(k3_ami_2) & v1_membered(k3_ami_2) & v2_membered(k3_ami_2) & v3_membered(k3_ami_2) & v4_membered(k3_ami_2) & v5_membered(k3_ami_2) ), file(ami_2,fc2_ami_2), [interesting(0.9),axiom,file(ami_2,fc2_ami_2)]). fof(fc2_ami_3,theorem, ( ~ v3_struct_0(k1_ami_3) & v1_ami_1(k1_ami_3,k1_tarski(k4_numbers)) & ~ v2_ami_1(k1_ami_3,k1_tarski(k4_numbers)) & v5_ami_1(k1_ami_3,k1_tarski(k4_numbers)) & v6_ami_1(k1_ami_3,k1_tarski(k4_numbers)) & v8_ami_1(k1_ami_3,k1_tarski(k4_numbers)) ), file(ami_3,fc2_ami_3), [interesting(0.9),axiom,file(ami_3,fc2_ami_3)]). fof(fc3_ami_3,theorem, ( ~ v3_struct_0(k1_ami_3) & v1_ami_1(k1_ami_3,k1_tarski(k4_numbers)) & ~ v2_ami_1(k1_ami_3,k1_tarski(k4_numbers)) & v4_ami_1(k1_ami_3,k1_tarski(k4_numbers)) & v5_ami_1(k1_ami_3,k1_tarski(k4_numbers)) & v6_ami_1(k1_ami_3,k1_tarski(k4_numbers)) & v8_ami_1(k1_ami_3,k1_tarski(k4_numbers)) ), file(ami_3,fc3_ami_3), [interesting(0.9),axiom,file(ami_3,fc3_ami_3)]). fof(fc4_numbers,theorem,( ~ v1_xboole_0(k4_numbers) ), file(numbers,fc4_numbers), [interesting(0.9),axiom,file(numbers,fc4_numbers)]). fof(fc5_ami_3,theorem, ( ~ v3_struct_0(k1_ami_3) & v1_ami_1(k1_ami_3,k1_tarski(k4_numbers)) & ~ v2_ami_1(k1_ami_3,k1_tarski(k4_numbers)) & v4_ami_1(k1_ami_3,k1_tarski(k4_numbers)) & v5_ami_1(k1_ami_3,k1_tarski(k4_numbers)) & v6_ami_1(k1_ami_3,k1_tarski(k4_numbers)) & v7_ami_1(k1_ami_3,k1_tarski(k4_numbers)) & v8_ami_1(k1_ami_3,k1_tarski(k4_numbers)) & v10_ami_1(k1_ami_3,k1_tarski(k4_numbers)) ), file(ami_3,fc5_ami_3), [interesting(0.9),axiom,file(ami_3,fc5_ami_3)]). fof(cc1_finseq_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) ) ), file(finseq_1,cc1_finseq_1), [interesting(0.9),axiom,file(finseq_1,cc1_finseq_1)]). fof(fc2_fraenkel,theorem,( ! [A,B] : ( ( v1_finset_1(A) & v1_finset_1(B) ) => ( v1_finset_1(k1_funct_2(A,B)) & v1_fraenkel(k1_funct_2(A,B)) ) ) ), file(fraenkel,fc2_fraenkel), [interesting(0.9),axiom,file(fraenkel,fc2_fraenkel)]). fof(rc1_finseq_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc1_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc1_finseq_1)]). fof(rc2_finseq_1,theorem,( ! [A] : ? [B] : ( m1_relset_1(B,k5_numbers,A) & v1_relat_1(B) & v1_funct_1(B) & v1_finset_1(B) & v1_finseq_1(B) ) ), file(finseq_1,rc2_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc2_finseq_1)]). fof(rc3_finseq_1,theorem,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc3_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc3_finseq_1)]). fof(rc4_funct_1,theorem,( ? [A] : ( v1_relat_1(A) & v3_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc4_funct_1), [interesting(0.9),axiom,file(funct_1,rc4_funct_1)]). fof(rc6_finseq_1,theorem,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) ) ), file(finseq_1,rc6_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc6_finseq_1)]). fof(rc7_finseq_1,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) ), file(finseq_1,rc7_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc7_finseq_1)]). fof(rc8_finseq_1,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc8_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc8_finseq_1)]). fof(commutativity_k2_xcmplx_0,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k2_xcmplx_0(A,B) = k2_xcmplx_0(B,A) ) ), file(xcmplx_0,k2_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k2_xcmplx_0)]). fof(commutativity_k3_xcmplx_0,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k3_xcmplx_0(A,B) = k3_xcmplx_0(B,A) ) ), file(xcmplx_0,k3_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k3_xcmplx_0)]). fof(existence_m1_fraenkel,axiom,( ! [A,B] : ? [C] : m1_fraenkel(C,A,B) ), file(fraenkel,m1_fraenkel), [interesting(0.9),axiom,file(fraenkel,m1_fraenkel)]). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]). fof(dt_k2_xcmplx_0,axiom,( $true ), file(xcmplx_0,k2_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k2_xcmplx_0)]). fof(dt_k3_xcmplx_0,axiom,( $true ), file(xcmplx_0,k3_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k3_xcmplx_0)]). fof(dt_m1_fraenkel,axiom,( ! [A,B,C] : ( m1_fraenkel(C,A,B) => ( ~ v1_xboole_0(C) & v1_fraenkel(C) ) ) ), file(fraenkel,m1_fraenkel), [interesting(0.9),axiom,file(fraenkel,m1_fraenkel)]). fof(fc1_int_1,theorem,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v1_int_1(k2_xcmplx_0(A,B)) ) ) ), file(int_1,fc1_int_1), [interesting(0.9),axiom,file(int_1,fc1_int_1)]). fof(fc1_xboole_0,theorem,( v1_xboole_0(k1_xboole_0) ), file(xboole_0,fc1_xboole_0), [interesting(0.9),axiom,file(xboole_0,fc1_xboole_0)]). fof(fc2_finseq_1,theorem, ( 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) ), file(finseq_1,fc2_finseq_1), [interesting(0.9),axiom,file(finseq_1,fc2_finseq_1)]). fof(fc2_int_1,theorem,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & v1_int_1(k3_xcmplx_0(A,B)) ) ) ), file(int_1,fc2_int_1), [interesting(0.9),axiom,file(int_1,fc2_int_1)]). fof(fc6_int_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_int_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v1_int_1(k2_xcmplx_0(B,A)) ) ) ), file(int_1,fc6_int_1), [interesting(0.9),axiom,file(int_1,fc6_int_1)]). fof(fc7_int_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_int_1(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(B,A)) & v1_xreal_0(k3_xcmplx_0(B,A)) & v1_int_1(k3_xcmplx_0(B,A)) ) ) ), file(int_1,fc7_int_1), [interesting(0.9),axiom,file(int_1,fc7_int_1)]). fof(rqRealAdd__k2_xcmplx_0__r0_r0_r0,theorem,( k2_xcmplx_0(0,0) = 0 ), file(arithm,rqRealAdd__k2_xcmplx_0__r0_r0_r0), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r0_r0_r0)]). fof(rqRealAdd__k2_xcmplx_0__r0_r1_r1,theorem,( k2_xcmplx_0(0,1) = 1 ), file(arithm,rqRealAdd__k2_xcmplx_0__r0_r1_r1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r0_r1_r1)]). fof(rqRealAdd__k2_xcmplx_0__r0_r2_r2,theorem,( k2_xcmplx_0(0,2) = 2 ), file(arithm,rqRealAdd__k2_xcmplx_0__r0_r2_r2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r0_r2_r2)]). fof(rqRealAdd__k2_xcmplx_0__r0_r9_r9,theorem,( k2_xcmplx_0(0,9) = 9 ), file(arithm,rqRealAdd__k2_xcmplx_0__r0_r9_r9), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r0_r9_r9)]). fof(rqRealAdd__k2_xcmplx_0__r1_r0_r1,theorem,( k2_xcmplx_0(1,0) = 1 ), file(arithm,rqRealAdd__k2_xcmplx_0__r1_r0_r1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r1_r0_r1)]). fof(rqRealAdd__k2_xcmplx_0__r1_r1_r2,theorem,( k2_xcmplx_0(1,1) = 2 ), file(arithm,rqRealAdd__k2_xcmplx_0__r1_r1_r2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r1_r1_r2)]). fof(rqRealAdd__k2_xcmplx_0__r2_r0_r2,theorem,( k2_xcmplx_0(2,0) = 2 ), file(arithm,rqRealAdd__k2_xcmplx_0__r2_r0_r2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r2_r0_r2)]). fof(rqRealAdd__k2_xcmplx_0__r9_r0_r9,theorem,( k2_xcmplx_0(9,0) = 9 ), file(arithm,rqRealAdd__k2_xcmplx_0__r9_r0_r9), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r9_r0_r9)]). fof(rqRealMult__k3_xcmplx_0__r0_r0_r0,theorem,( k3_xcmplx_0(0,0) = 0 ), file(arithm,rqRealMult__k3_xcmplx_0__r0_r0_r0), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r0_r0_r0)]). fof(rqRealMult__k3_xcmplx_0__r0_r1_r0,theorem,( k3_xcmplx_0(0,1) = 0 ), file(arithm,rqRealMult__k3_xcmplx_0__r0_r1_r0), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r0_r1_r0)]). fof(rqRealMult__k3_xcmplx_0__r0_r2_r0,theorem,( k3_xcmplx_0(0,2) = 0 ), file(arithm,rqRealMult__k3_xcmplx_0__r0_r2_r0), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r0_r2_r0)]). fof(rqRealMult__k3_xcmplx_0__r1_r0_r0,theorem,( k3_xcmplx_0(1,0) = 0 ), file(arithm,rqRealMult__k3_xcmplx_0__r1_r0_r0), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r1_r0_r0)]). fof(rqRealMult__k3_xcmplx_0__r1_r1_r1,theorem,( k3_xcmplx_0(1,1) = 1 ), file(arithm,rqRealMult__k3_xcmplx_0__r1_r1_r1), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r1_r1_r1)]). fof(rqRealMult__k3_xcmplx_0__r1_r2_r2,theorem,( k3_xcmplx_0(1,2) = 2 ), file(arithm,rqRealMult__k3_xcmplx_0__r1_r2_r2), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r1_r2_r2)]). fof(rqRealMult__k3_xcmplx_0__r1_r9_r9,theorem,( k3_xcmplx_0(1,9) = 9 ), file(arithm,rqRealMult__k3_xcmplx_0__r1_r9_r9), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r1_r9_r9)]). fof(rqRealMult__k3_xcmplx_0__r2_r0_r0,theorem,( k3_xcmplx_0(2,0) = 0 ), file(arithm,rqRealMult__k3_xcmplx_0__r2_r0_r0), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r2_r0_r0)]). fof(rqRealMult__k3_xcmplx_0__r2_r1_r2,theorem,( k3_xcmplx_0(2,1) = 2 ), file(arithm,rqRealMult__k3_xcmplx_0__r2_r1_r2), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r2_r1_r2)]). fof(rqRealMult__k3_xcmplx_0__r9_r1_r9,theorem,( k3_xcmplx_0(9,1) = 9 ), file(arithm,rqRealMult__k3_xcmplx_0__r9_r1_r9), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r9_r1_r9)]). fof(spc5_arithm,theorem,( ! [A,B,C] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) & v1_xcmplx_0(C) ) => k3_xcmplx_0(k2_xcmplx_0(A,B),C) = k2_xcmplx_0(k3_xcmplx_0(A,C),k3_xcmplx_0(B,C)) ) ), file(arithm,spc5_arithm), [interesting(0.9),axiom,file(arithm,spc5_arithm)]). fof(spc6_arithm,theorem,( ! [A,B,C] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) & v1_xcmplx_0(C) ) => k2_xcmplx_0(k2_xcmplx_0(A,B),C) = k2_xcmplx_0(A,k2_xcmplx_0(B,C)) ) ), file(arithm,spc6_arithm), [interesting(0.9),axiom,file(arithm,spc6_arithm)]). fof(spc7_arithm,theorem,( ! [A,B,C] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) & v1_xcmplx_0(C) ) => k3_xcmplx_0(k3_xcmplx_0(A,B),C) = k3_xcmplx_0(A,k3_xcmplx_0(B,C)) ) ), file(arithm,spc6_arithm), [interesting(0.9),axiom,file(arithm,spc6_arithm)]). fof(t1_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k2_xcmplx_0(A,0) = A ) ), file(arithm,t1_arithm), [interesting(0.9),axiom,file(arithm,t1_arithm)]). fof(t1_boole,theorem,( ! [A] : k2_xboole_0(A,k1_xboole_0) = A ), file(boole,t1_boole), [interesting(0.9),axiom,file(boole,t1_boole)]). fof(t1_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & v2_xreal_0(A) ) => v2_xreal_0(B) ) ) ) ), file(real,t1_real), [interesting(0.9),axiom,file(real,t1_real)]). fof(t2_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(A,0) = 0 ) ), file(arithm,t2_arithm), [interesting(0.9),axiom,file(arithm,t2_arithm)]). fof(t2_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & v3_xreal_0(B) ) => v3_xreal_0(A) ) ) ) ), file(real,t2_real), [interesting(0.9),axiom,file(real,t2_real)]). fof(t3_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(1,A) = A ) ), file(arithm,t3_arithm), [interesting(0.9),axiom,file(arithm,t3_arithm)]). fof(t3_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( r1_xreal_0(A,B) & ~ v3_xreal_0(A) & v3_xreal_0(B) ) ) ) ), file(real,t3_real), [interesting(0.9),axiom,file(real,t3_real)]). fof(t4_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( r1_xreal_0(A,B) & ~ v2_xreal_0(B) & v2_xreal_0(A) ) ) ) ), file(real,t4_real), [interesting(0.9),axiom,file(real,t4_real)]). fof(t5_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(B) | v3_xreal_0(A) | v2_xreal_0(B) ) ) ) ) ), file(real,t5_real), [interesting(0.9),axiom,file(real,t5_real)]). fof(t6_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(A) | v2_xreal_0(B) | v3_xreal_0(A) ) ) ) ) ), file(real,t6_real), [interesting(0.9),axiom,file(real,t6_real)]). fof(t7_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,B) & ~ v2_xreal_0(A) & ~ v3_xreal_0(B) ) ) ) ), file(real,t7_real), [interesting(0.9),axiom,file(real,t7_real)]). fof(t8_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,B) & ~ v3_xreal_0(B) & ~ v2_xreal_0(A) ) ) ) ), file(real,t8_real), [interesting(0.9),axiom,file(real,t8_real)]). fof(commutativity_k1_nat_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k1_nat_1(A,B) = k1_nat_1(B,A) ) ), file(nat_1,k1_nat_1), [interesting(0.9),axiom,file(nat_1,k1_nat_1)]). fof(commutativity_k2_nat_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k2_nat_1(A,B) = k2_nat_1(B,A) ) ), file(nat_1,k2_nat_1), [interesting(0.9),axiom,file(nat_1,k2_nat_1)]). fof(reflexivity_r1_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => r1_xreal_0(A,A) ) ), file(xreal_0,r1_xreal_0), [interesting(0.9),axiom,file(xreal_0,r1_xreal_0)]). fof(connectedness_r1_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( r1_xreal_0(A,B) | r1_xreal_0(B,A) ) ) ), file(xreal_0,r1_xreal_0), [interesting(0.9),axiom,file(xreal_0,r1_xreal_0)]). fof(antisymmetry_r2_hidden,theorem,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), [interesting(0.9),axiom,file(hidden,r2_hidden)]). fof(existence_l1_ami_1,axiom,( ! [A] : ? [B] : l1_ami_1(B,A) ), file(ami_1,l1_ami_1), [interesting(0.9),axiom,file(ami_1,l1_ami_1)]). fof(existence_l1_struct_0,axiom,( ? [A] : l1_struct_0(A) ), file(struct_0,l1_struct_0), [interesting(0.9),axiom,file(struct_0,l1_struct_0)]). fof(existence_m1_relset_1,axiom,( ! [A,B] : ? [C] : m1_relset_1(C,A,B) ), file(relset_1,m1_relset_1), [interesting(0.9),axiom,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), [interesting(0.9),axiom,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), [interesting(0.9),axiom,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), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(redefinition_k1_fraenkel,definition,( ! [A,B] : ( ~ v1_xboole_0(B) => k1_fraenkel(A,B) = k1_funct_2(A,B) ) ), file(fraenkel,k1_fraenkel), [interesting(0.9),axiom,file(fraenkel,k1_fraenkel)]). fof(redefinition_k1_nat_1,definition,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k1_nat_1(A,B) = k2_xcmplx_0(A,B) ) ), file(nat_1,k1_nat_1), [interesting(0.9),axiom,file(nat_1,k1_nat_1)]). fof(redefinition_k2_nat_1,definition,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k2_nat_1(A,B) = k3_xcmplx_0(A,B) ) ), file(nat_1,k2_nat_1), [interesting(0.9),axiom,file(nat_1,k2_nat_1)]). fof(redefinition_m2_subset_1,definition,( ! [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), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(dt_k1_fraenkel,axiom,( ! [A,B] : ( ~ v1_xboole_0(B) => m1_fraenkel(k1_fraenkel(A,B),A,B) ) ), file(fraenkel,k1_fraenkel), [interesting(0.9),axiom,file(fraenkel,k1_fraenkel)]). fof(dt_k1_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k1_nat_1(A,B),k1_numbers,k5_numbers) ) ), file(nat_1,k1_nat_1), [interesting(0.9),axiom,file(nat_1,k1_nat_1)]). fof(dt_k2_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k2_nat_1(A,B),k1_numbers,k5_numbers) ) ), file(nat_1,k2_nat_1), [interesting(0.9),axiom,file(nat_1,k2_nat_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), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(rqLessOrEqual__r1_xreal_0__r0_r0,theorem,( r1_xreal_0(0,0) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_r0)]). fof(rqLessOrEqual__r1_xreal_0__r0_r1,theorem,( r1_xreal_0(0,1) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_r1)]). fof(rqLessOrEqual__r1_xreal_0__r0_r2,theorem,( r1_xreal_0(0,2) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_r2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_r2)]). fof(rqLessOrEqual__r1_xreal_0__r0_r9,theorem,( r1_xreal_0(0,9) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_r9), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_r9)]). fof(rqLessOrEqual__r1_xreal_0__r1_r0,theorem,( ~ r1_xreal_0(1,0) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_r0)]). fof(rqLessOrEqual__r1_xreal_0__r1_r1,theorem,( r1_xreal_0(1,1) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_r1)]). fof(rqLessOrEqual__r1_xreal_0__r1_r2,theorem,( r1_xreal_0(1,2) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_r2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_r2)]). fof(rqLessOrEqual__r1_xreal_0__r1_r9,theorem,( r1_xreal_0(1,9) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_r9), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_r9)]). fof(rqLessOrEqual__r1_xreal_0__r2_r0,theorem,( ~ r1_xreal_0(2,0) ), file(arithm,rqLessOrEqual__r1_xreal_0__r2_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r2_r0)]). fof(rqLessOrEqual__r1_xreal_0__r2_r1,theorem,( ~ r1_xreal_0(2,1) ), file(arithm,rqLessOrEqual__r1_xreal_0__r2_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r2_r1)]). fof(rqLessOrEqual__r1_xreal_0__r2_r2,theorem,( r1_xreal_0(2,2) ), file(arithm,rqLessOrEqual__r1_xreal_0__r2_r2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r2_r2)]). fof(rqLessOrEqual__r1_xreal_0__r2_r9,theorem,( r1_xreal_0(2,9) ), file(arithm,rqLessOrEqual__r1_xreal_0__r2_r9), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r2_r9)]). fof(rqLessOrEqual__r1_xreal_0__r9_r0,theorem,( ~ r1_xreal_0(9,0) ), file(arithm,rqLessOrEqual__r1_xreal_0__r9_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r9_r0)]). fof(rqLessOrEqual__r1_xreal_0__r9_r1,theorem,( ~ r1_xreal_0(9,1) ), file(arithm,rqLessOrEqual__r1_xreal_0__r9_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r9_r1)]). fof(rqLessOrEqual__r1_xreal_0__r9_r2,theorem,( ~ r1_xreal_0(9,2) ), file(arithm,rqLessOrEqual__r1_xreal_0__r9_r2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r9_r2)]). fof(rqLessOrEqual__r1_xreal_0__r9_r9,theorem,( r1_xreal_0(9,9) ), file(arithm,rqLessOrEqual__r1_xreal_0__r9_r9), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r9_r9)]). fof(spc1_boole,theorem,( ~ v1_xboole_0(1) ), file(boole,spc1_boole), [interesting(0.9),axiom,file(boole,spc1_boole)]). fof(spc1_numerals,theorem, ( v2_xreal_0(1) & m2_subset_1(1,k1_numbers,k5_numbers) & m1_subset_1(1,k5_numbers) & m1_subset_1(1,k1_numbers) ), file(numerals,spc1_numerals), [interesting(0.9),axiom,file(numerals,spc1_numerals)]). fof(spc2_boole,theorem,( ~ v1_xboole_0(2) ), file(boole,spc2_boole), [interesting(0.9),axiom,file(boole,spc2_boole)]). fof(spc2_numerals,theorem, ( v2_xreal_0(2) & m2_subset_1(2,k1_numbers,k5_numbers) & m1_subset_1(2,k5_numbers) & m1_subset_1(2,k1_numbers) ), file(numerals,spc2_numerals), [interesting(0.9),axiom,file(numerals,spc2_numerals)]). fof(t1_numerals,theorem,( m1_subset_1(0,k5_numbers) ), file(numerals,t1_numerals), [interesting(0.9),axiom,file(numerals,t1_numerals)]). fof(t1_subset,theorem,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), [interesting(0.9),axiom,file(subset,t1_subset)]). fof(t2_subset,theorem,( ! [A,B] : ( m1_subset_1(A,B) => ( v1_xboole_0(B) | r2_hidden(A,B) ) ) ), file(subset,t2_subset), [interesting(0.9),axiom,file(subset,t2_subset)]). fof(t4_subset,theorem,( ! [A,B,C] : ( ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) ) => m1_subset_1(A,C) ) ), file(subset,t4_subset), [interesting(0.9),axiom,file(subset,t4_subset)]). fof(t5_subset,theorem,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), [interesting(0.9),axiom,file(subset,t5_subset)]). fof(t6_boole,theorem,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), [interesting(0.9),axiom,file(boole,t6_boole)]). fof(t7_boole,theorem,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), [interesting(0.9),axiom,file(boole,t7_boole)]). fof(t8_boole,theorem,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), [interesting(0.9),axiom,file(boole,t8_boole)]). fof(reflexivity_r1_tarski,theorem,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), [interesting(0.9),axiom,file(tarski,r1_tarski)]). fof(redefinition_k6_domain_1,definition,( ! [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), [interesting(0.9),axiom,file(domain_1,k6_domain_1)]). fof(dt_k17_ami_2,axiom, ( v1_funct_1(k17_ami_2) & v1_funct_2(k17_ami_2,k4_ami_2,k1_fraenkel(k4_card_3(k5_ami_2),k4_card_3(k5_ami_2))) & m2_relset_1(k17_ami_2,k4_ami_2,k1_fraenkel(k4_card_3(k5_ami_2),k4_card_3(k5_ami_2))) ), file(ami_2,k17_ami_2), [interesting(0.9),axiom,file(ami_2,k17_ami_2)]). fof(dt_k1_gr_cy_1,axiom,( ! [A] : ( v4_ordinal2(A) => ( ~ v1_xboole_0(k1_gr_cy_1(A)) & m1_subset_1(k1_gr_cy_1(A),k1_zfmisc_1(k5_numbers)) ) ) ), file(gr_cy_1,k1_gr_cy_1), [interesting(0.9),axiom,file(gr_cy_1,k1_gr_cy_1)]). fof(dt_k4_ami_2,axiom,( m1_subset_1(k4_ami_2,k1_zfmisc_1(k2_zfmisc_1(k1_gr_cy_1(9),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(k4_numbers)),k5_numbers))))) ), file(ami_2,k4_ami_2), [interesting(0.9),axiom,file(ami_2,k4_ami_2)]). fof(dt_k5_ami_2,axiom, ( v1_funct_1(k5_ami_2) & v1_funct_2(k5_ami_2,k5_numbers,k2_xboole_0(k1_tarski(k4_numbers),k2_tarski(k4_ami_2,k3_ami_2))) & m2_relset_1(k5_ami_2,k5_numbers,k2_xboole_0(k1_tarski(k4_numbers),k2_tarski(k4_ami_2,k3_ami_2))) ), file(ami_2,k5_ami_2), [interesting(0.9),axiom,file(ami_2,k5_ami_2)]). 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), [interesting(0.9),axiom,file(domain_1,k6_domain_1)]). fof(fc3_ami_2,theorem, ( v1_relat_1(k4_ami_2) & ~ v1_xboole_0(k4_ami_2) ), file(ami_2,fc3_ami_2), [interesting(0.9),axiom,file(ami_2,fc3_ami_2)]). fof(t3_subset,theorem,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), [interesting(0.9),axiom,file(subset,t3_subset)]). fof(t2_tarski,theorem,( ! [A,B] : ( ! [C] : ( r2_hidden(C,A) <=> r2_hidden(C,B) ) => A = B ) ), file(tarski,t2_tarski), [interesting(0.9),axiom,file(tarski,t2_tarski)]). fof(fraenkel_a_0_0_ami_2,definition,( ! [A] : ( r2_hidden(A,a_0_0_ami_2) <=> ? [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) & A = k1_nat_1(k2_nat_1(2,B),1) ) ) ), file(ami_2,a_0_0_ami_2), [interesting(0.9),axiom,file(ami_2,a_0_0_ami_2)]). fof(fraenkel_a_0_0_ami_5,definition,( ! [A] : ( r2_hidden(A,a_0_0_ami_5) <=> ? [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) & A = k1_nat_1(k2_nat_1(2,B),1) ) ) ), file(ami_5,a_0_0_ami_5), [interesting(0.9),axiom,file(ami_5,a_0_0_ami_5)]). fof(fraenkel_a_0_1_ami_2,definition,( ! [A] : ( r2_hidden(A,a_0_1_ami_2) <=> ? [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) & A = k2_nat_1(2,B) & ~ r1_xreal_0(B,0) ) ) ), file(ami_2,a_0_1_ami_2), [interesting(0.9),axiom,file(ami_2,a_0_1_ami_2)]). fof(fraenkel_a_0_1_ami_5,definition,( ! [A] : ( r2_hidden(A,a_0_1_ami_5) <=> ? [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) & A = k2_nat_1(2,B) & ~ r1_xreal_0(B,0) ) ) ), file(ami_5,a_0_1_ami_5), [interesting(0.9),axiom,file(ami_5,a_0_1_ami_5)]). fof(d5_ami_1,definition,( ! [A,B] : ( ( ~ v3_struct_0(B) & l1_ami_1(B,A) ) => k2_ami_1(A,B) = u1_ami_1(A,B) ) ), file(ami_1,d5_ami_1), [interesting(0.9),axiom,file(ami_1,d5_ami_1)]). fof(spc0_numerals,theorem, ( v2_xreal_0(0) & m2_subset_1(0,k1_numbers,k5_numbers) & m1_subset_1(0,k5_numbers) & m1_subset_1(0,k1_numbers) ), file(numerals,spc0_numerals), [interesting(0.9),axiom,file(numerals,spc0_numerals)]). fof(spc9_numerals,theorem, ( v2_xreal_0(9) & m2_subset_1(9,k1_numbers,k5_numbers) & m1_subset_1(9,k5_numbers) & m1_subset_1(9,k1_numbers) ), file(numerals,spc9_numerals), [interesting(0.9),axiom,file(numerals,spc9_numerals)]). fof(spc0_boole,theorem,( v1_xboole_0(0) ), file(boole,spc0_boole), [interesting(0.9),axiom,file(boole,spc0_boole)]). fof(spc9_boole,theorem,( ~ v1_xboole_0(9) ), file(boole,spc9_boole), [interesting(0.9),axiom,file(boole,spc9_boole)]). fof(d1_ami_3,definition,( k1_ami_3 = g1_ami_1(k1_tarski(k4_numbers),k5_numbers,0,k3_ami_2,k1_gr_cy_1(9),k4_ami_2,k5_ami_2,k17_ami_2) ), file(ami_3,d1_ami_3), [interesting(0.9),axiom,file(ami_3,d1_ami_3)]). fof(t4_ami_3,theorem,( k2_ami_1(k1_tarski(k4_numbers),k1_ami_3) = 0 ), file(ami_3,t4_ami_3), [interesting(0.9),axiom,file(ami_3,t4_ami_3)]). fof(t37_zfmisc_1,theorem,( ! [A,B] : ( r1_tarski(k1_tarski(A),B) <=> r2_hidden(A,B) ) ), file(zfmisc_1,t37_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,t37_zfmisc_1)]). fof(e2_5__ami_5,plain,( r1_tarski(k1_struct_0(k1_ami_3,k2_ami_1(k1_tarski(k4_numbers),k1_ami_3)),k5_numbers) ), inference(mizar_by,[status(thm),assumptions([])],[existence_m1_fraenkel,dt_m1_fraenkel,cc1_finseq_1,cc1_fraenkel,fc2_fraenkel,rc1_finseq_1,rc1_fraenkel,rc2_finseq_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,commutativity_k2_tarski,commutativity_k2_xboole_0,idempotence_k2_xboole_0,existence_m1_relset_1,existence_m2_relset_1,redefinition_k1_fraenkel,redefinition_m2_relset_1,dt_k13_finseq_1,dt_k1_fraenkel,dt_k1_funct_2,dt_k1_xboole_0,dt_k2_tarski,dt_k2_xboole_0,dt_k2_zfmisc_1,dt_k3_tarski,dt_k4_card_3,dt_m1_relset_1,dt_m2_relset_1,dt_u2_ami_1,dt_u3_ami_1,dt_u4_ami_1,dt_u5_ami_1,dt_u6_ami_1,cc1_arytm_3,cc1_relset_1,cc1_setfam_1,cc2_arytm_3,cc2_funct_1,cc3_int_1,cc4_int_1,fc16_finseq_1,fc1_fraenkel,fc1_xboole_0,fc2_ami_1,fc2_finseq_1,fc2_xboole_0,fc3_ami_1,fc3_setfam_1,fc3_xboole_0,fc4_ami_1,fc4_setfam_1,fc9_finseq_1,rc1_arytm_3,rc1_funct_1,rc1_int_1,rc1_setfam_1,rc2_funct_1,rc2_int_1,rc3_ami_1,rc7_ami_1,t1_boole,free_g1_ami_1,abstractness_v1_ami_1,existence_l1_ami_1,existence_l1_struct_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_m2_subset_1,dt_g1_ami_1,dt_k17_ami_2,dt_k1_gr_cy_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k3_ami_2,dt_k4_ami_2,dt_k5_ami_2,dt_k5_ordinal2,dt_l1_ami_1,dt_l1_struct_0,dt_m1_subset_1,dt_m2_subset_1,dt_u1_ami_1,dt_u1_struct_0,cc1_funct_1,cc1_int_1,cc2_int_1,cc3_arytm_3,fc1_numbers,fc1_ordinal2,fc1_struct_0,fc2_ami_2,fc2_setfam_1,fc3_ami_2,rc1_ami_1,rc1_xboole_0,rc2_ami_1,rc2_xboole_0,rc3_struct_0,rc5_ami_1,rc5_struct_0,rc6_ami_1,rc8_ami_1,rc9_ami_1,t1_numerals,t2_subset,t4_subset,t5_subset,t6_boole,t8_boole,spc9_numerals,spc9_boole,reflexivity_r1_tarski,antisymmetry_r2_hidden,redefinition_k1_struct_0,redefinition_k5_numbers,dt_k1_ami_3,dt_k1_struct_0,dt_k1_tarski,dt_k2_ami_1,dt_k4_numbers,dt_k5_numbers,fc1_ami_3,fc2_ami_3,fc3_ami_3,fc4_numbers,fc5_ami_3,t1_subset,t3_subset,t7_boole,d1_ami_3,d5_ami_1,spc0_numerals,spc0_boole,t4_ami_3,t37_zfmisc_1]), [interesting(0.8),file(ami_5,e2_5__ami_5),[file(ami_5,e2_5__ami_5)]]). fof(t8_xboole_1,theorem,( ! [A,B,C] : ( ( r1_tarski(A,B) & r1_tarski(C,B) ) => r1_tarski(k2_xboole_0(A,C),B) ) ), file(xboole_1,t8_xboole_1), [interesting(0.9),axiom,file(xboole_1,t8_xboole_1)]). fof(e3_5__ami_5,plain,( r1_tarski(k2_xboole_0(k1_struct_0(k1_ami_3,k2_ami_1(k1_tarski(k4_numbers),k1_ami_3)),k2_ami_2),k5_numbers) ), inference(mizar_by,[status(thm),assumptions([])],[existence_m1_fraenkel,dt_m1_fraenkel,cc1_finseq_1,cc1_fraenkel,fc2_fraenkel,rc1_finseq_1,rc1_fraenkel,rc2_finseq_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,commutativity_k2_tarski,antisymmetry_r2_hidden,existence_m1_relset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k1_fraenkel,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k13_finseq_1,dt_k1_fraenkel,dt_k1_funct_2,dt_k1_xboole_0,dt_k2_tarski,dt_k2_zfmisc_1,dt_k3_tarski,dt_k4_card_3,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u2_ami_1,dt_u3_ami_1,dt_u4_ami_1,dt_u5_ami_1,dt_u6_ami_1,cc1_arytm_3,cc1_relset_1,cc1_setfam_1,cc2_arytm_3,cc2_funct_1,cc3_int_1,cc4_int_1,fc16_finseq_1,fc1_fraenkel,fc1_xboole_0,fc2_ami_1,fc2_finseq_1,fc3_ami_1,fc3_setfam_1,fc4_ami_1,fc4_setfam_1,fc9_finseq_1,rc1_arytm_3,rc1_funct_1,rc1_int_1,rc1_setfam_1,rc2_funct_1,rc2_int_1,rc3_ami_1,rc7_ami_1,t1_boole,t1_subset,t4_subset,t5_subset,free_g1_ami_1,abstractness_v1_ami_1,existence_l1_ami_1,existence_l1_struct_0,existence_m1_subset_1,dt_g1_ami_1,dt_k17_ami_2,dt_k1_gr_cy_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k3_ami_2,dt_k4_ami_2,dt_k5_ami_2,dt_k5_ordinal2,dt_l1_ami_1,dt_l1_struct_0,dt_m1_subset_1,dt_u1_ami_1,dt_u1_struct_0,cc1_funct_1,cc1_int_1,cc2_int_1,cc3_arytm_3,fc1_numbers,fc1_ordinal2,fc1_struct_0,fc2_ami_2,fc2_setfam_1,fc2_xboole_0,fc3_ami_2,fc3_xboole_0,rc1_ami_1,rc1_xboole_0,rc2_ami_1,rc2_xboole_0,rc3_struct_0,rc5_ami_1,rc5_struct_0,rc6_ami_1,rc8_ami_1,rc9_ami_1,spc0_boole,t1_numerals,t2_subset,t6_boole,t7_boole,t8_boole,spc0_numerals,spc9_numerals,spc0_boole,spc9_boole,commutativity_k2_xboole_0,idempotence_k2_xboole_0,reflexivity_r1_tarski,redefinition_k1_struct_0,redefinition_k5_numbers,dt_k1_ami_3,dt_k1_struct_0,dt_k1_tarski,dt_k2_ami_1,dt_k2_ami_2,dt_k2_xboole_0,dt_k4_numbers,dt_k5_numbers,fc1_ami_2,fc1_ami_3,fc2_ami_3,fc3_ami_3,fc4_numbers,fc5_ami_3,t3_subset,d1_ami_3,d5_ami_1,e2_5__ami_5,t8_xboole_1]), [interesting(0.8),file(ami_5,e3_5__ami_5),[file(ami_5,e3_5__ami_5)]]). fof(e4_5__ami_5,plain,( r1_tarski(k2_xboole_0(k2_xboole_0(k1_struct_0(k1_ami_3,k2_ami_1(k1_tarski(k4_numbers),k1_ami_3)),k2_ami_2),k3_ami_2),k5_numbers) ), inference(mizar_by,[status(thm),assumptions([])],[existence_m1_fraenkel,dt_m1_fraenkel,cc1_finseq_1,cc1_fraenkel,fc2_fraenkel,rc1_finseq_1,rc1_fraenkel,rc2_finseq_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,commutativity_k2_tarski,antisymmetry_r2_hidden,existence_m1_relset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k1_fraenkel,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k13_finseq_1,dt_k1_fraenkel,dt_k1_funct_2,dt_k1_xboole_0,dt_k2_tarski,dt_k2_zfmisc_1,dt_k3_tarski,dt_k4_card_3,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u2_ami_1,dt_u3_ami_1,dt_u4_ami_1,dt_u5_ami_1,dt_u6_ami_1,cc1_arytm_3,cc1_relset_1,cc1_setfam_1,cc2_arytm_3,cc2_funct_1,cc3_int_1,cc4_int_1,fc16_finseq_1,fc1_fraenkel,fc1_xboole_0,fc2_ami_1,fc2_finseq_1,fc3_ami_1,fc3_setfam_1,fc4_ami_1,fc4_setfam_1,fc9_finseq_1,rc1_arytm_3,rc1_funct_1,rc1_int_1,rc1_setfam_1,rc2_funct_1,rc2_int_1,rc3_ami_1,rc7_ami_1,t1_boole,t1_subset,t4_subset,t5_subset,free_g1_ami_1,abstractness_v1_ami_1,existence_l1_ami_1,existence_l1_struct_0,existence_m1_subset_1,dt_g1_ami_1,dt_k17_ami_2,dt_k1_gr_cy_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k4_ami_2,dt_k5_ami_2,dt_k5_ordinal2,dt_l1_ami_1,dt_l1_struct_0,dt_m1_subset_1,dt_u1_ami_1,dt_u1_struct_0,cc1_funct_1,cc1_int_1,cc2_int_1,cc3_arytm_3,fc1_numbers,fc1_ordinal2,fc1_struct_0,fc2_setfam_1,fc2_xboole_0,fc3_ami_2,fc3_xboole_0,rc1_ami_1,rc1_xboole_0,rc2_ami_1,rc2_xboole_0,rc3_struct_0,rc5_ami_1,rc5_struct_0,rc6_ami_1,rc8_ami_1,rc9_ami_1,spc0_boole,t1_numerals,t2_subset,t6_boole,t7_boole,t8_boole,spc0_numerals,spc9_numerals,spc0_boole,spc9_boole,commutativity_k2_xboole_0,idempotence_k2_xboole_0,reflexivity_r1_tarski,redefinition_k1_struct_0,redefinition_k5_numbers,dt_k1_ami_3,dt_k1_struct_0,dt_k1_tarski,dt_k2_ami_1,dt_k2_ami_2,dt_k2_xboole_0,dt_k3_ami_2,dt_k4_numbers,dt_k5_numbers,fc1_ami_2,fc1_ami_3,fc2_ami_2,fc2_ami_3,fc3_ami_3,fc4_numbers,fc5_ami_3,t3_subset,d1_ami_3,d5_ami_1,e3_5__ami_5,t8_xboole_1]), [interesting(0.8),file(ami_5,e4_5__ami_5),[file(ami_5,e4_5__ami_5)]]). fof(dt_c1_5_1__ami_5,assumption,( $true ), introduced(assumption,[file(ami_5,c1_5_1__ami_5)]), [interesting(0.65),axiom,file(ami_5,c1_5_1__ami_5)]). fof(d3_tarski,definition,( ! [A,B] : ( r1_tarski(A,B) <=> ! [C] : ( r2_hidden(C,A) => r2_hidden(C,B) ) ) ), file(tarski,d3_tarski), [interesting(0.9),axiom,file(tarski,d3_tarski)]). fof(dh_c1_5_1__ami_5,definition, ( ~ ( r2_hidden(c1_5_1__ami_5,k5_numbers) & ~ r2_hidden(c1_5_1__ami_5,k2_xboole_0(k2_xboole_0(k6_domain_1(k5_numbers,0),a_0_0_ami_5),a_0_1_ami_5)) ) => ! [A] : ~ ( r2_hidden(A,k5_numbers) & ~ r2_hidden(A,k2_xboole_0(k2_xboole_0(k6_domain_1(k5_numbers,0),a_0_0_ami_5),a_0_1_ami_5)) ) ), introduced(definition,[new_symbol(c1_5_1__ami_5),file(ami_5,c1_5_1__ami_5)]), [interesting(0.65),axiom,file(ami_5,c1_5_1__ami_5)]). fof(e1_5_1__ami_5,assumption,( r2_hidden(c1_5_1__ami_5,k5_numbers) ), introduced(assumption,[file(ami_5,e1_5_1__ami_5)]), [interesting(0.65),axiom,file(ami_5,e1_5_1__ami_5)]). fof(e1_5_1_1_1__ami_5,assumption,( c1_5_1__ami_5 = 0 ), introduced(assumption,[file(ami_5,e1_5_1_1_1__ami_5)]), [interesting(0.35),axiom,file(ami_5,e1_5_1_1_1__ami_5)]). fof(d1_tarski,definition,( ! [A,B] : ( B = k1_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> C = A ) ) ), file(tarski,d1_tarski), [interesting(0.9),axiom,file(tarski,d1_tarski)]). fof(e2_5_1_1_1__ami_5,plain,( r2_hidden(c1_5_1__ami_5,k6_domain_1(k5_numbers,0)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_5_1__ami_5,e1_5_1_1_1__ami_5])],[cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,cc1_arytm_3,cc1_setfam_1,cc2_arytm_3,cc3_int_1,cc4_int_1,fc1_xboole_0,fc2_finseq_1,rc1_arytm_3,rc1_int_1,rc1_setfam_1,rc2_int_1,existence_m1_subset_1,existence_m2_subset_1,redefinition_m2_subset_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,dt_m2_subset_1,cc1_funct_1,cc2_int_1,cc3_arytm_3,fc1_numbers,fc1_ordinal2,fc2_setfam_1,rc1_xboole_0,rc2_xboole_0,t1_numerals,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,redefinition_k5_numbers,redefinition_k6_domain_1,dt_k1_tarski,dt_k5_numbers,dt_k6_domain_1,dt_c1_5_1__ami_5,t1_subset,t7_boole,spc0_numerals,spc0_boole,e1_5_1_1_1__ami_5,d1_tarski]), [interesting(0.35),file(ami_5,e2_5_1_1_1__ami_5),[file(ami_5,e2_5_1_1_1__ami_5)]]). fof(d2_xboole_0,definition,( ! [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), [interesting(0.9),axiom,file(xboole_0,d2_xboole_0)]). fof(e3_5_1_1_1__ami_5,plain,( r2_hidden(c1_5_1__ami_5,k2_xboole_0(k6_domain_1(k5_numbers,0),a_0_0_ami_5)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_5_1__ami_5,e1_5_1_1_1__ami_5])],[cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,commutativity_k2_xcmplx_0,commutativity_k3_xcmplx_0,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k2_xcmplx_0,dt_k3_xcmplx_0,cc1_arytm_3,cc1_setfam_1,cc2_arytm_3,cc3_int_1,cc4_int_1,fc1_int_1,fc1_xboole_0,fc2_finseq_1,fc2_int_1,fc4_setfam_1,fc6_int_1,fc7_int_1,rc1_arytm_3,rc1_int_1,rc1_setfam_1,rc2_int_1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_r2_r2,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqRealAdd__k2_xcmplx_0__r2_r0_r2,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,spc5_arithm,spc6_arithm,spc7_arithm,t1_arithm,t1_boole,t2_arithm,t3_arithm,commutativity_k1_nat_1,commutativity_k2_nat_1,existence_m1_subset_1,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k2_nat_1,redefinition_m2_subset_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_tarski,dt_k1_zfmisc_1,dt_k2_nat_1,dt_k5_ordinal2,dt_m1_subset_1,dt_m2_subset_1,cc1_funct_1,cc2_int_1,cc3_arytm_3,fc1_numbers,fc1_ordinal2,fc2_setfam_1,fc2_xboole_0,fc3_xboole_0,rc1_xboole_0,rc2_xboole_0,spc1_boole,spc1_numerals,spc2_boole,spc2_numerals,t1_numerals,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,spc1_numerals,spc2_numerals,spc1_boole,spc2_boole,commutativity_k2_xboole_0,idempotence_k2_xboole_0,antisymmetry_r2_hidden,redefinition_k5_numbers,redefinition_k6_domain_1,dt_k2_xboole_0,dt_k5_numbers,dt_k6_domain_1,dt_c1_5_1__ami_5,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_0_ami_5,spc0_numerals,spc0_boole,e2_5_1_1_1__ami_5,d2_xboole_0]), [interesting(0.35),file(ami_5,e3_5_1_1_1__ami_5),[file(ami_5,e3_5_1_1_1__ami_5)]]). fof(e4_5_1_1_1__ami_5,plain,( r2_hidden(c1_5_1__ami_5,k2_xboole_0(k2_xboole_0(k6_domain_1(k5_numbers,0),a_0_0_ami_5),a_0_1_ami_5)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_5_1__ami_5,e1_5_1_1_1__ami_5])],[cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,commutativity_k2_xcmplx_0,commutativity_k3_xcmplx_0,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k2_xcmplx_0,dt_k3_xcmplx_0,cc1_arytm_3,cc1_setfam_1,cc2_arytm_3,cc3_int_1,cc4_int_1,fc1_int_1,fc1_xboole_0,fc2_finseq_1,fc2_int_1,fc4_setfam_1,fc6_int_1,fc7_int_1,rc1_arytm_3,rc1_int_1,rc1_setfam_1,rc2_int_1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_r2_r2,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqRealAdd__k2_xcmplx_0__r2_r0_r2,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,spc5_arithm,spc6_arithm,spc7_arithm,t1_arithm,t1_boole,t1_real,t2_arithm,t2_real,t3_arithm,t3_real,t4_real,t5_real,t6_real,t7_real,t8_real,commutativity_k1_nat_1,commutativity_k2_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k2_nat_1,redefinition_m2_subset_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_tarski,dt_k1_zfmisc_1,dt_k2_nat_1,dt_k5_ordinal2,dt_m1_subset_1,dt_m2_subset_1,cc1_funct_1,cc2_int_1,cc3_arytm_3,fc1_numbers,fc1_ordinal2,fc2_setfam_1,fc2_xboole_0,fc3_xboole_0,rc1_xboole_0,rc2_xboole_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,spc1_boole,spc1_numerals,spc2_boole,spc2_numerals,t1_numerals,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,spc1_numerals,spc2_numerals,spc1_boole,spc2_boole,commutativity_k2_xboole_0,idempotence_k2_xboole_0,antisymmetry_r2_hidden,redefinition_k5_numbers,redefinition_k6_domain_1,dt_k2_xboole_0,dt_k5_numbers,dt_k6_domain_1,dt_c1_5_1__ami_5,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_0_ami_5,fraenkel_a_0_1_ami_5,spc0_numerals,spc0_boole,e3_5_1_1_1__ami_5,d2_xboole_0]), [interesting(0.35),file(ami_5,e4_5_1_1_1__ami_5),[file(ami_5,e4_5_1_1_1__ami_5)]]). fof(i2_5_1_1_1__ami_5,theorem,( $true ), introduced(tautology,[file(ami_5,i2_5_1_1_1__ami_5)]), [interesting(0.35),trivial,file(ami_5,i2_5_1_1_1__ami_5)]). fof(i1_5_1_1_1__ami_5,plain,( r2_hidden(c1_5_1__ami_5,k2_xboole_0(k2_xboole_0(k6_domain_1(k5_numbers,0),a_0_0_ami_5),a_0_1_ami_5)) ), inference(conclusion,[status(thm),assumptions([dt_c1_5_1__ami_5,e1_5_1_1_1__ami_5])],[e4_5_1_1_1__ami_5,i2_5_1_1_1__ami_5]), [interesting(0.35),file(ami_5,i1_5_1_1_1__ami_5),[file(ami_5,i1_5_1_1_1__ami_5)]]). fof(i1_5_1_1__ami_5,plain, ( c1_5_1__ami_5 = 0 => r2_hidden(c1_5_1__ami_5,k2_xboole_0(k2_xboole_0(k6_domain_1(k5_numbers,0),a_0_0_ami_5),a_0_1_ami_5)) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_5_1__ami_5]),discharge_asm(discharge,[e1_5_1_1_1__ami_5])],[e1_5_1_1_1__ami_5,i1_5_1_1_1__ami_5]), [interesting(0.5),file(ami_5,i1_5_1_1__ami_5),[file(ami_5,i1_5_1_1__ami_5)]]). fof(e1_5_1_1_2__ami_5,assumption,( ? [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) & c1_5_1__ami_5 = k1_nat_1(k2_nat_1(2,A),1) ) ), introduced(assumption,[file(ami_5,e1_5_1_1_2__ami_5)]), [interesting(0.35),axiom,file(ami_5,e1_5_1_1_2__ami_5)]). fof(fc3_int_1,theorem,( ! [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), [interesting(0.9),axiom,file(int_1,fc3_int_1)]). fof(fc4_int_1,theorem,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v1_int_1(k6_xcmplx_0(A,B)) ) ) ), file(int_1,fc4_int_1), [interesting(0.9),axiom,file(int_1,fc4_int_1)]). fof(fc8_int_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_int_1(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & v1_int_1(k6_xcmplx_0(B,A)) ) ) ), file(int_1,fc8_int_1), [interesting(0.9),axiom,file(int_1,fc8_int_1)]). fof(fc5_int_1,theorem,( ! [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), [interesting(0.9),axiom,file(int_1,fc5_int_1)]). fof(fc9_int_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v1_int_1(k6_xcmplx_0(A,B)) ) ) ), file(int_1,fc9_int_1), [interesting(0.9),axiom,file(int_1,fc9_int_1)]). fof(rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,theorem,( k2_xcmplx_0(0,k4_xcmplx_0(1)) = k4_xcmplx_0(1) ), file(arithm,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1)]). fof(rqRealAdd__k2_xcmplx_0__r0_rm2_rm2,theorem,( k2_xcmplx_0(0,k4_xcmplx_0(2)) = k4_xcmplx_0(2) ), file(arithm,rqRealAdd__k2_xcmplx_0__r0_rm2_rm2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r0_rm2_rm2)]). fof(rqRealAdd__k2_xcmplx_0__r1_rm1_r0,theorem,( k2_xcmplx_0(1,k4_xcmplx_0(1)) = 0 ), file(arithm,rqRealAdd__k2_xcmplx_0__r1_rm1_r0), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r1_rm1_r0)]). fof(rqRealAdd__k2_xcmplx_0__r1_rm2_rm1,theorem,( k2_xcmplx_0(1,k4_xcmplx_0(2)) = k4_xcmplx_0(1) ), file(arithm,rqRealAdd__k2_xcmplx_0__r1_rm2_rm1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r1_rm2_rm1)]). fof(rqRealAdd__k2_xcmplx_0__r2_rm1_r1,theorem,( k2_xcmplx_0(2,k4_xcmplx_0(1)) = 1 ), file(arithm,rqRealAdd__k2_xcmplx_0__r2_rm1_r1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r2_rm1_r1)]). fof(rqRealAdd__k2_xcmplx_0__r2_rm2_r0,theorem,( k2_xcmplx_0(2,k4_xcmplx_0(2)) = 0 ), file(arithm,rqRealAdd__k2_xcmplx_0__r2_rm2_r0), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r2_rm2_r0)]). fof(rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,theorem,( k2_xcmplx_0(k4_xcmplx_0(1),0) = k4_xcmplx_0(1) ), file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1)]). fof(rqRealAdd__k2_xcmplx_0__rm1_r1_r0,theorem,( k2_xcmplx_0(k4_xcmplx_0(1),1) = 0 ), file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r1_r0), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r1_r0)]). fof(rqRealAdd__k2_xcmplx_0__rm1_r2_r1,theorem,( k2_xcmplx_0(k4_xcmplx_0(1),2) = 1 ), file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r2_r1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r2_r1)]). fof(rqRealAdd__k2_xcmplx_0__rm1_rm1_rm2,theorem,( k2_xcmplx_0(k4_xcmplx_0(1),k4_xcmplx_0(1)) = k4_xcmplx_0(2) ), file(arithm,rqRealAdd__k2_xcmplx_0__rm1_rm1_rm2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm1_rm1_rm2)]). fof(rqRealAdd__k2_xcmplx_0__rm2_r0_rm2,theorem,( k2_xcmplx_0(k4_xcmplx_0(2),0) = k4_xcmplx_0(2) ), file(arithm,rqRealAdd__k2_xcmplx_0__rm2_r0_rm2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm2_r0_rm2)]). fof(rqRealAdd__k2_xcmplx_0__rm2_r1_rm1,theorem,( k2_xcmplx_0(k4_xcmplx_0(2),1) = k4_xcmplx_0(1) ), file(arithm,rqRealAdd__k2_xcmplx_0__rm2_r1_rm1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm2_r1_rm1)]). fof(rqRealAdd__k2_xcmplx_0__rm2_r2_r0,theorem,( k2_xcmplx_0(k4_xcmplx_0(2),2) = 0 ), file(arithm,rqRealAdd__k2_xcmplx_0__rm2_r2_r0), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm2_r2_r0)]). fof(spc1_arithm,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k2_xcmplx_0(A,k4_xcmplx_0(B)) = k6_xcmplx_0(A,B) ) ), file(arithm,spc1_arithm), [interesting(0.9),axiom,file(arithm,spc1_arithm)]). fof(spc2_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(A,k4_xcmplx_0(1)) = k4_xcmplx_0(A) ) ), file(arithm,spc2_arithm), [interesting(0.9),axiom,file(arithm,spc2_arithm)]). fof(spc8_arithm,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k2_xcmplx_0(k4_xcmplx_0(A),k4_xcmplx_0(B)) = k4_xcmplx_0(k2_xcmplx_0(A,B)) ) ), file(arithm,spc8_arithm), [interesting(0.9),axiom,file(arithm,spc8_arithm)]). fof(spc9_arithm,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k6_xcmplx_0(k4_xcmplx_0(A),k4_xcmplx_0(B)) = k6_xcmplx_0(B,A) ) ), file(arithm,spc9_arithm), [interesting(0.9),axiom,file(arithm,spc9_arithm)]). fof(t4_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k6_xcmplx_0(A,0) = A ) ), file(arithm,t4_arithm), [interesting(0.9),axiom,file(arithm,t4_arithm)]). fof(involutiveness_k4_xcmplx_0,theorem,( ! [A] : ( v1_xcmplx_0(A) => k4_xcmplx_0(k4_xcmplx_0(A)) = A ) ), file(xcmplx_0,k4_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k4_xcmplx_0)]). fof(dt_k4_xcmplx_0,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_xcmplx_0(k4_xcmplx_0(A)) ) ), file(xcmplx_0,k4_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k4_xcmplx_0)]). fof(dt_k6_xcmplx_0,axiom,( $true ), file(xcmplx_0,k6_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k6_xcmplx_0)]). fof(rqRealDiff__k6_xcmplx_0__r0_r0_r0,theorem,( k6_xcmplx_0(0,0) = 0 ), file(arithm,rqRealDiff__k6_xcmplx_0__r0_r0_r0), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r0_r0_r0)]). fof(rqRealDiff__k6_xcmplx_0__r0_r1_rm1,theorem,( k6_xcmplx_0(0,1) = k4_xcmplx_0(1) ), file(arithm,rqRealDiff__k6_xcmplx_0__r0_r1_rm1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r0_r1_rm1)]). fof(rqRealDiff__k6_xcmplx_0__r0_r2_rm2,theorem,( k6_xcmplx_0(0,2) = k4_xcmplx_0(2) ), file(arithm,rqRealDiff__k6_xcmplx_0__r0_r2_rm2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r0_r2_rm2)]). fof(rqRealDiff__k6_xcmplx_0__r0_rm1_r1,theorem,( k6_xcmplx_0(0,k4_xcmplx_0(1)) = 1 ), file(arithm,rqRealDiff__k6_xcmplx_0__r0_rm1_r1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r0_rm1_r1)]). fof(rqRealDiff__k6_xcmplx_0__r0_rm2_r2,theorem,( k6_xcmplx_0(0,k4_xcmplx_0(2)) = 2 ), file(arithm,rqRealDiff__k6_xcmplx_0__r0_rm2_r2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r0_rm2_r2)]). fof(rqRealDiff__k6_xcmplx_0__r1_r0_r1,theorem,( k6_xcmplx_0(1,0) = 1 ), file(arithm,rqRealDiff__k6_xcmplx_0__r1_r0_r1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r1_r0_r1)]). fof(rqRealDiff__k6_xcmplx_0__r1_rm1_r2,theorem,( k6_xcmplx_0(1,k4_xcmplx_0(1)) = 2 ), file(arithm,rqRealDiff__k6_xcmplx_0__r1_rm1_r2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r1_rm1_r2)]). fof(rqRealDiff__k6_xcmplx_0__r2_r0_r2,theorem,( k6_xcmplx_0(2,0) = 2 ), file(arithm,rqRealDiff__k6_xcmplx_0__r2_r0_r2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r2_r0_r2)]). fof(rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,theorem,( k6_xcmplx_0(k4_xcmplx_0(1),0) = k4_xcmplx_0(1) ), file(arithm,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1)]). fof(rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,theorem,( k6_xcmplx_0(k4_xcmplx_0(1),1) = k4_xcmplx_0(2) ), file(arithm,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2)]). fof(rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,theorem,( k6_xcmplx_0(k4_xcmplx_0(2),0) = k4_xcmplx_0(2) ), file(arithm,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2)]). fof(rqRealMult__k3_xcmplx_0__r0_rm2_r0,theorem,( k3_xcmplx_0(0,k4_xcmplx_0(2)) = 0 ), file(arithm,rqRealMult__k3_xcmplx_0__r0_rm2_r0), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r0_rm2_r0)]). fof(rqRealMult__k3_xcmplx_0__r1_rm2_rm2,theorem,( k3_xcmplx_0(1,k4_xcmplx_0(2)) = k4_xcmplx_0(2) ), file(arithm,rqRealMult__k3_xcmplx_0__r1_rm2_rm2), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r1_rm2_rm2)]). fof(rqRealMult__k3_xcmplx_0__rm2_r0_r0,theorem,( k3_xcmplx_0(k4_xcmplx_0(2),0) = 0 ), file(arithm,rqRealMult__k3_xcmplx_0__rm2_r0_r0), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__rm2_r0_r0)]). fof(rqRealMult__k3_xcmplx_0__rm2_r1_rm2,theorem,( k3_xcmplx_0(k4_xcmplx_0(2),1) = k4_xcmplx_0(2) ), file(arithm,rqRealMult__k3_xcmplx_0__rm2_r1_rm2), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__rm2_r1_rm2)]). fof(rqRealNeg__k4_xcmplx_0__r0_r0,theorem,( k4_xcmplx_0(0) = 0 ), file(arithm,rqRealNeg__k4_xcmplx_0__r0_r0), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__r0_r0)]). fof(rqRealNeg__k4_xcmplx_0__r2_rm2,theorem,( k4_xcmplx_0(2) = k4_xcmplx_0(2) ), file(arithm,rqRealNeg__k4_xcmplx_0__r2_rm2), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__r2_rm2)]). fof(rqRealNeg__k4_xcmplx_0__r1_rm1,theorem,( k4_xcmplx_0(1) = k4_xcmplx_0(1) ), file(arithm,rqRealNeg__k4_xcmplx_0__r1_rm1), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__r1_rm1)]). fof(rqRealDiff__k6_xcmplx_0__r1_r1_r0,theorem,( k6_xcmplx_0(1,1) = 0 ), file(arithm,rqRealDiff__k6_xcmplx_0__r1_r1_r0), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r1_r1_r0)]). fof(rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,theorem,( k6_xcmplx_0(k4_xcmplx_0(2),k4_xcmplx_0(2)) = 0 ), file(arithm,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0)]). fof(rqRealNeg__k4_xcmplx_0__rm1_r1,theorem,( k4_xcmplx_0(k4_xcmplx_0(1)) = 1 ), file(arithm,rqRealNeg__k4_xcmplx_0__rm1_r1), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__rm1_r1)]). fof(rqRealNeg__k4_xcmplx_0__rm2_r2,theorem,( k4_xcmplx_0(k4_xcmplx_0(2)) = 2 ), file(arithm,rqRealNeg__k4_xcmplx_0__rm2_r2), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__rm2_r2)]). fof(rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,theorem,( k6_xcmplx_0(k4_xcmplx_0(1),k4_xcmplx_0(1)) = 0 ), file(arithm,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0)]). fof(rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,theorem,( k6_xcmplx_0(k4_xcmplx_0(2),k4_xcmplx_0(1)) = k4_xcmplx_0(1) ), file(arithm,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1)]). fof(rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,theorem,( k6_xcmplx_0(k4_xcmplx_0(1),k4_xcmplx_0(2)) = 1 ), file(arithm,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1)]). fof(rqRealDiff__k6_xcmplx_0__r2_r2_r0,theorem,( k6_xcmplx_0(2,2) = 0 ), file(arithm,rqRealDiff__k6_xcmplx_0__r2_r2_r0), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r2_r2_r0)]). fof(rqRealDiff__k6_xcmplx_0__r1_r2_rm1,theorem,( k6_xcmplx_0(1,2) = k4_xcmplx_0(1) ), file(arithm,rqRealDiff__k6_xcmplx_0__r1_r2_rm1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r1_r2_rm1)]). fof(rqRealDiff__k6_xcmplx_0__r2_r1_r1,theorem,( k6_xcmplx_0(2,1) = 1 ), file(arithm,rqRealDiff__k6_xcmplx_0__r2_r1_r1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r2_r1_r1)]). fof(e2_5_1_1_2__ami_5,plain,( r2_hidden(c1_5_1__ami_5,a_0_0_ami_5) ), inference(mizar_by,[status(thm),assumptions([dt_c1_5_1__ami_5,e1_5_1_1_2__ami_5])],[cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,cc1_arytm_3,cc2_arytm_3,cc3_int_1,cc4_int_1,fc1_int_1,fc1_xboole_0,fc2_finseq_1,fc2_int_1,fc3_int_1,fc4_int_1,fc6_int_1,fc7_int_1,fc8_int_1,rc1_arytm_3,rc1_int_1,rc2_int_1,commutativity_k2_xcmplx_0,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_subset_1,cc1_funct_1,cc2_int_1,cc3_arytm_3,fc1_ordinal2,fc5_int_1,fc9_int_1,rc1_xboole_0,rc2_xboole_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_r2_r2,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,rqRealAdd__k2_xcmplx_0__r0_rm2_rm2,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__r1_rm2_rm1,rqRealAdd__k2_xcmplx_0__r2_r0_r2,rqRealAdd__k2_xcmplx_0__r2_rm1_r1,rqRealAdd__k2_xcmplx_0__r2_rm2_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0,rqRealAdd__k2_xcmplx_0__rm1_r2_r1,rqRealAdd__k2_xcmplx_0__rm1_rm1_rm2,rqRealAdd__k2_xcmplx_0__rm2_r0_rm2,rqRealAdd__k2_xcmplx_0__rm2_r1_rm1,rqRealAdd__k2_xcmplx_0__rm2_r2_r0,spc1_arithm,spc2_arithm,spc5_arithm,spc6_arithm,spc7_arithm,spc8_arithm,spc9_arithm,t1_arithm,t1_numerals,t2_arithm,t2_subset,t3_arithm,t3_subset,t4_arithm,t4_subset,t5_subset,t6_boole,t8_boole,commutativity_k1_nat_1,commutativity_k2_nat_1,commutativity_k3_xcmplx_0,involutiveness_k4_xcmplx_0,antisymmetry_r2_hidden,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k2_nat_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_nat_1,dt_k1_numbers,dt_k2_nat_1,dt_k3_xcmplx_0,dt_k4_xcmplx_0,dt_k5_numbers,dt_k6_xcmplx_0,dt_m2_subset_1,dt_c1_5_1__ami_5,fc1_numbers,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r0_rm2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_rm2_rm2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,rqRealMult__k3_xcmplx_0__rm2_r0_r0,rqRealMult__k3_xcmplx_0__rm2_r1_rm2,rqRealNeg__k4_xcmplx_0__r0_r0,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_0_ami_5,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e1_5_1_1_2__ami_5,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r2_r1_r1]), [interesting(0.35),file(ami_5,e2_5_1_1_2__ami_5),[file(ami_5,e2_5_1_1_2__ami_5)]]). fof(e3_5_1_1_2__ami_5,plain,( r2_hidden(c1_5_1__ami_5,k2_xboole_0(k6_domain_1(k5_numbers,0),a_0_0_ami_5)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_5_1__ami_5,e1_5_1_1_2__ami_5])],[cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,commutativity_k2_xcmplx_0,commutativity_k3_xcmplx_0,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k2_xcmplx_0,dt_k3_xcmplx_0,cc1_arytm_3,cc1_setfam_1,cc2_arytm_3,cc3_int_1,cc4_int_1,fc1_int_1,fc1_xboole_0,fc2_finseq_1,fc2_int_1,fc4_setfam_1,fc6_int_1,fc7_int_1,rc1_arytm_3,rc1_int_1,rc1_setfam_1,rc2_int_1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_r2_r2,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqRealAdd__k2_xcmplx_0__r2_r0_r2,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,spc5_arithm,spc6_arithm,spc7_arithm,t1_arithm,t1_boole,t2_arithm,t3_arithm,commutativity_k1_nat_1,commutativity_k2_nat_1,existence_m1_subset_1,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k2_nat_1,redefinition_m2_subset_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_tarski,dt_k1_zfmisc_1,dt_k2_nat_1,dt_k5_ordinal2,dt_m1_subset_1,dt_m2_subset_1,cc1_funct_1,cc2_int_1,cc3_arytm_3,fc1_numbers,fc1_ordinal2,fc2_setfam_1,fc2_xboole_0,fc3_xboole_0,rc1_xboole_0,rc2_xboole_0,spc1_boole,spc1_numerals,spc2_boole,spc2_numerals,t1_numerals,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,spc1_numerals,spc2_numerals,spc1_boole,spc2_boole,commutativity_k2_xboole_0,idempotence_k2_xboole_0,antisymmetry_r2_hidden,redefinition_k5_numbers,redefinition_k6_domain_1,dt_k2_xboole_0,dt_k5_numbers,dt_k6_domain_1,dt_c1_5_1__ami_5,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_0_ami_5,spc0_numerals,spc0_boole,e2_5_1_1_2__ami_5,d2_xboole_0]), [interesting(0.35),file(ami_5,e3_5_1_1_2__ami_5),[file(ami_5,e3_5_1_1_2__ami_5)]]). fof(e4_5_1_1_2__ami_5,plain,( r2_hidden(c1_5_1__ami_5,k2_xboole_0(k2_xboole_0(k6_domain_1(k5_numbers,0),a_0_0_ami_5),a_0_1_ami_5)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_5_1__ami_5,e1_5_1_1_2__ami_5])],[cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,commutativity_k2_xcmplx_0,commutativity_k3_xcmplx_0,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k2_xcmplx_0,dt_k3_xcmplx_0,cc1_arytm_3,cc1_setfam_1,cc2_arytm_3,cc3_int_1,cc4_int_1,fc1_int_1,fc1_xboole_0,fc2_finseq_1,fc2_int_1,fc4_setfam_1,fc6_int_1,fc7_int_1,rc1_arytm_3,rc1_int_1,rc1_setfam_1,rc2_int_1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_r2_r2,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqRealAdd__k2_xcmplx_0__r2_r0_r2,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,spc5_arithm,spc6_arithm,spc7_arithm,t1_arithm,t1_boole,t1_real,t2_arithm,t2_real,t3_arithm,t3_real,t4_real,t5_real,t6_real,t7_real,t8_real,commutativity_k1_nat_1,commutativity_k2_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k2_nat_1,redefinition_m2_subset_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_tarski,dt_k1_zfmisc_1,dt_k2_nat_1,dt_k5_ordinal2,dt_m1_subset_1,dt_m2_subset_1,cc1_funct_1,cc2_int_1,cc3_arytm_3,fc1_numbers,fc1_ordinal2,fc2_setfam_1,fc2_xboole_0,fc3_xboole_0,rc1_xboole_0,rc2_xboole_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,spc1_boole,spc1_numerals,spc2_boole,spc2_numerals,t1_numerals,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,spc1_numerals,spc2_numerals,spc1_boole,spc2_boole,commutativity_k2_xboole_0,idempotence_k2_xboole_0,antisymmetry_r2_hidden,redefinition_k5_numbers,redefinition_k6_domain_1,dt_k2_xboole_0,dt_k5_numbers,dt_k6_domain_1,dt_c1_5_1__ami_5,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_0_ami_5,fraenkel_a_0_1_ami_5,spc0_numerals,spc0_boole,e3_5_1_1_2__ami_5,d2_xboole_0]), [interesting(0.35),file(ami_5,e4_5_1_1_2__ami_5),[file(ami_5,e4_5_1_1_2__ami_5)]]). fof(i2_5_1_1_2__ami_5,theorem,( $true ), introduced(tautology,[file(ami_5,i2_5_1_1_2__ami_5)]), [interesting(0.35),trivial,file(ami_5,i2_5_1_1_2__ami_5)]). fof(i1_5_1_1_2__ami_5,plain,( r2_hidden(c1_5_1__ami_5,k2_xboole_0(k2_xboole_0(k6_domain_1(k5_numbers,0),a_0_0_ami_5),a_0_1_ami_5)) ), inference(conclusion,[status(thm),assumptions([dt_c1_5_1__ami_5,e1_5_1_1_2__ami_5])],[e4_5_1_1_2__ami_5,i2_5_1_1_2__ami_5]), [interesting(0.35),file(ami_5,i1_5_1_1_2__ami_5),[file(ami_5,i1_5_1_1_2__ami_5)]]). fof(i2_5_1_1__ami_5,plain, ( ? [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) & c1_5_1__ami_5 = k1_nat_1(k2_nat_1(2,A),1) ) => r2_hidden(c1_5_1__ami_5,k2_xboole_0(k2_xboole_0(k6_domain_1(k5_numbers,0),a_0_0_ami_5),a_0_1_ami_5)) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_5_1__ami_5]),discharge_asm(discharge,[e1_5_1_1_2__ami_5])],[e1_5_1_1_2__ami_5,i1_5_1_1_2__ami_5]), [interesting(0.5),file(ami_5,i2_5_1_1__ami_5),[file(ami_5,i2_5_1_1__ami_5)]]). fof(e1_5_1_1_3__ami_5,assumption,( ? [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) & c1_5_1__ami_5 = k2_nat_1(2,A) & ~ r1_xreal_0(A,0) ) ), introduced(assumption,[file(ami_5,e1_5_1_1_3__ami_5)]), [interesting(0.35),axiom,file(ami_5,e1_5_1_1_3__ami_5)]). fof(rqLessOrEqual__r1_xreal_0__r0_rm1,theorem,( ~ r1_xreal_0(0,k4_xcmplx_0(1)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_rm1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_rm1)]). fof(rqLessOrEqual__r1_xreal_0__r0_rm2,theorem,( ~ r1_xreal_0(0,k4_xcmplx_0(2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_rm2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_rm2)]). fof(rqLessOrEqual__r1_xreal_0__r1_rm1,theorem,( ~ r1_xreal_0(1,k4_xcmplx_0(1)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_rm1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_rm1)]). fof(rqLessOrEqual__r1_xreal_0__r1_rm2,theorem,( ~ r1_xreal_0(1,k4_xcmplx_0(2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_rm2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_rm2)]). fof(rqLessOrEqual__r1_xreal_0__r2_rm1,theorem,( ~ r1_xreal_0(2,k4_xcmplx_0(1)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r2_rm1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r2_rm1)]). fof(rqLessOrEqual__r1_xreal_0__r2_rm2,theorem,( ~ r1_xreal_0(2,k4_xcmplx_0(2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r2_rm2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r2_rm2)]). fof(rqLessOrEqual__r1_xreal_0__rm1_r0,theorem,( r1_xreal_0(k4_xcmplx_0(1),0) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r0)]). fof(rqLessOrEqual__r1_xreal_0__rm1_r1,theorem,( r1_xreal_0(k4_xcmplx_0(1),1) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r1)]). fof(rqLessOrEqual__r1_xreal_0__rm1_r2,theorem,( r1_xreal_0(k4_xcmplx_0(1),2) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r2)]). fof(rqLessOrEqual__r1_xreal_0__rm1_rm1,theorem,( r1_xreal_0(k4_xcmplx_0(1),k4_xcmplx_0(1)) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm1_rm1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm1_rm1)]). fof(rqLessOrEqual__r1_xreal_0__rm1_rm2,theorem,( ~ r1_xreal_0(k4_xcmplx_0(1),k4_xcmplx_0(2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm1_rm2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm1_rm2)]). fof(rqLessOrEqual__r1_xreal_0__rm2_r0,theorem,( r1_xreal_0(k4_xcmplx_0(2),0) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm2_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm2_r0)]). fof(rqLessOrEqual__r1_xreal_0__rm2_r1,theorem,( r1_xreal_0(k4_xcmplx_0(2),1) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm2_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm2_r1)]). fof(rqLessOrEqual__r1_xreal_0__rm2_r2,theorem,( r1_xreal_0(k4_xcmplx_0(2),2) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm2_r2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm2_r2)]). fof(rqLessOrEqual__r1_xreal_0__rm2_rm1,theorem,( r1_xreal_0(k4_xcmplx_0(2),k4_xcmplx_0(1)) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm2_rm1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm2_rm1)]). fof(rqLessOrEqual__r1_xreal_0__rm2_rm2,theorem,( r1_xreal_0(k4_xcmplx_0(2),k4_xcmplx_0(2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm2_rm2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm2_rm2)]). fof(e2_5_1_1_3__ami_5,plain,( r2_hidden(c1_5_1__ami_5,a_0_1_ami_5) ), inference(mizar_by,[status(thm),assumptions([dt_c1_5_1__ami_5,e1_5_1_1_3__ami_5])],[cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,cc1_arytm_3,cc2_arytm_3,cc3_int_1,cc4_int_1,fc1_xboole_0,fc2_finseq_1,fc2_int_1,fc3_int_1,fc4_int_1,fc7_int_1,fc8_int_1,rc1_arytm_3,rc1_int_1,rc2_int_1,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc1_funct_1,cc2_int_1,cc3_arytm_3,fc1_ordinal2,fc5_int_1,fc9_int_1,rc1_xboole_0,rc2_xboole_0,spc2_arithm,spc7_arithm,spc9_arithm,t1_numerals,t1_real,t2_arithm,t2_subset,t3_arithm,t3_subset,t4_arithm,t4_real,t4_subset,t5_subset,t6_boole,t8_boole,commutativity_k2_nat_1,commutativity_k3_xcmplx_0,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m2_subset_1,redefinition_k2_nat_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_nat_1,dt_k3_xcmplx_0,dt_k4_xcmplx_0,dt_k5_numbers,dt_k6_xcmplx_0,dt_m2_subset_1,dt_c1_5_1__ami_5,fc1_numbers,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r0_rm2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r1_rm2,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r2_rm1,rqLessOrEqual__r1_xreal_0__r2_rm2,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_r2,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqLessOrEqual__r1_xreal_0__rm1_rm2,rqLessOrEqual__r1_xreal_0__rm2_r0,rqLessOrEqual__r1_xreal_0__rm2_r1,rqLessOrEqual__r1_xreal_0__rm2_r2,rqLessOrEqual__r1_xreal_0__rm2_rm1,rqLessOrEqual__r1_xreal_0__rm2_rm2,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r0_rm2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_rm2_rm2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,rqRealMult__k3_xcmplx_0__rm2_r0_r0,rqRealMult__k3_xcmplx_0__rm2_r1_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_1_ami_5,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e1_5_1_1_3__ami_5,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__rm2_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqLessOrEqual__r1_xreal_0__r2_r0]), [interesting(0.35),file(ami_5,e2_5_1_1_3__ami_5),[file(ami_5,e2_5_1_1_3__ami_5)]]). fof(e3_5_1_1_3__ami_5,plain,( r2_hidden(c1_5_1__ami_5,k2_xboole_0(k2_xboole_0(k6_domain_1(k5_numbers,0),a_0_0_ami_5),a_0_1_ami_5)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_5_1__ami_5,e1_5_1_1_3__ami_5])],[cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,commutativity_k2_xcmplx_0,commutativity_k3_xcmplx_0,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k2_xcmplx_0,dt_k3_xcmplx_0,cc1_arytm_3,cc1_setfam_1,cc2_arytm_3,cc3_int_1,cc4_int_1,fc1_int_1,fc1_xboole_0,fc2_finseq_1,fc2_int_1,fc4_setfam_1,fc6_int_1,fc7_int_1,rc1_arytm_3,rc1_int_1,rc1_setfam_1,rc2_int_1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_r2_r2,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqRealAdd__k2_xcmplx_0__r2_r0_r2,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,spc5_arithm,spc6_arithm,spc7_arithm,t1_arithm,t1_boole,t1_real,t2_arithm,t2_real,t3_arithm,t3_real,t4_real,t5_real,t6_real,t7_real,t8_real,commutativity_k1_nat_1,commutativity_k2_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k2_nat_1,redefinition_m2_subset_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_tarski,dt_k1_zfmisc_1,dt_k2_nat_1,dt_k5_ordinal2,dt_m1_subset_1,dt_m2_subset_1,cc1_funct_1,cc2_int_1,cc3_arytm_3,fc1_numbers,fc1_ordinal2,fc2_setfam_1,fc2_xboole_0,fc3_xboole_0,rc1_xboole_0,rc2_xboole_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,spc1_boole,spc1_numerals,spc2_boole,spc2_numerals,t1_numerals,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,spc1_numerals,spc2_numerals,spc1_boole,spc2_boole,commutativity_k2_xboole_0,idempotence_k2_xboole_0,antisymmetry_r2_hidden,redefinition_k5_numbers,redefinition_k6_domain_1,dt_k2_xboole_0,dt_k5_numbers,dt_k6_domain_1,dt_c1_5_1__ami_5,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_0_ami_5,fraenkel_a_0_1_ami_5,spc0_numerals,spc0_boole,e2_5_1_1_3__ami_5,d2_xboole_0]), [interesting(0.35),file(ami_5,e3_5_1_1_3__ami_5),[file(ami_5,e3_5_1_1_3__ami_5)]]). fof(i2_5_1_1_3__ami_5,theorem,( $true ), introduced(tautology,[file(ami_5,i2_5_1_1_3__ami_5)]), [interesting(0.35),trivial,file(ami_5,i2_5_1_1_3__ami_5)]). fof(i1_5_1_1_3__ami_5,plain,( r2_hidden(c1_5_1__ami_5,k2_xboole_0(k2_xboole_0(k6_domain_1(k5_numbers,0),a_0_0_ami_5),a_0_1_ami_5)) ), inference(conclusion,[status(thm),assumptions([dt_c1_5_1__ami_5,e1_5_1_1_3__ami_5])],[e3_5_1_1_3__ami_5,i2_5_1_1_3__ami_5]), [interesting(0.35),file(ami_5,i1_5_1_1_3__ami_5),[file(ami_5,i1_5_1_1_3__ami_5)]]). fof(i3_5_1_1__ami_5,plain, ( ? [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) & c1_5_1__ami_5 = k2_nat_1(2,A) & ~ r1_xreal_0(A,0) ) => r2_hidden(c1_5_1__ami_5,k2_xboole_0(k2_xboole_0(k6_domain_1(k5_numbers,0),a_0_0_ami_5),a_0_1_ami_5)) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_5_1__ami_5]),discharge_asm(discharge,[e1_5_1_1_3__ami_5])],[e1_5_1_1_3__ami_5,i1_5_1_1_3__ami_5]), [interesting(0.5),file(ami_5,i3_5_1_1__ami_5),[file(ami_5,i3_5_1_1__ami_5)]]). fof(dt_k3_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => m2_subset_1(k3_nat_1(A,B),k1_numbers,k5_numbers) ) ), file(nat_1,k3_nat_1), [interesting(0.9),axiom,file(nat_1,k3_nat_1)]). fof(de_c2_5_1__ami_5,definition,( c2_5_1__ami_5 = c1_5_1__ami_5 ), introduced(definition,[new_symbol(c2_5_1__ami_5),file(ami_5,c2_5_1__ami_5)]), [interesting(0.65),axiom,file(ami_5,c2_5_1__ami_5)]). fof(e2_5_1__ami_5,plain,( m2_subset_1(c1_5_1__ami_5,k1_numbers,k5_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_5_1__ami_5,e1_5_1__ami_5])],[cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,cc1_arytm_3,cc2_arytm_3,cc3_int_1,cc4_int_1,fc1_xboole_0,fc2_finseq_1,rc1_arytm_3,rc1_int_1,rc2_int_1,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc1_funct_1,cc2_int_1,cc3_arytm_3,fc1_ordinal2,rc1_xboole_0,rc2_xboole_0,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m2_subset_1,dt_c1_5_1__ami_5,fc1_numbers,t1_subset,t7_boole,e1_5_1__ami_5]), [interesting(0.65),file(ami_5,e2_5_1__ami_5),[file(ami_5,e2_5_1__ami_5)]]). fof(dt_c2_5_1__ami_5,plain,( m2_subset_1(c2_5_1__ami_5,k1_numbers,k5_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_5_1__ami_5,e1_5_1__ami_5])],[cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,cc1_arytm_3,cc2_arytm_3,cc3_int_1,cc4_int_1,fc1_xboole_0,fc2_finseq_1,rc1_arytm_3,rc1_int_1,rc2_int_1,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc1_funct_1,cc2_int_1,cc3_arytm_3,fc1_ordinal2,rc1_xboole_0,rc2_xboole_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m2_subset_1,dt_c1_5_1__ami_5,fc1_numbers,de_c2_5_1__ami_5,e2_5_1__ami_5]), [interesting(0.65),file(ami_5,c2_5_1__ami_5),[file(ami_5,c2_5_1__ami_5)]]). fof(t19_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => ~ ( 0 != A & r1_xreal_0(A,0) ) ) ), file(nat_1,t19_nat_1), [interesting(0.9),axiom,file(nat_1,t19_nat_1)]). fof(e3_5_1__ami_5,plain,( ~ ( k3_nat_1(c2_5_1__ami_5,2) != 0 & r1_xreal_0(k3_nat_1(c2_5_1__ami_5,2),0) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_5_1__ami_5,e1_5_1__ami_5])],[reflexivity_r1_tarski,cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc1_arytm_3,cc2_arytm_3,cc3_arytm_3,fc1_ordinal2,fc1_xboole_0,fc2_finseq_1,rc1_arytm_3,rc1_int_1,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_c1_5_1__ami_5,cc1_funct_1,cc2_int_1,cc4_int_1,fc1_numbers,rc1_xboole_0,rc2_int_1,rc2_xboole_0,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,dt_k3_nat_1,dt_c2_5_1__ami_5,de_c2_5_1__ami_5,cc3_int_1,rqLessOrEqual__r1_xreal_0__r2_r0,spc0_numerals,spc2_numerals,spc0_boole,spc2_boole,t19_nat_1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r2_r2]), [interesting(0.65),file(ami_5,e3_5_1__ami_5),[file(ami_5,e3_5_1__ami_5)]]). fof(dt_k4_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => m2_subset_1(k4_nat_1(A,B),k1_numbers,k5_numbers) ) ), file(nat_1,k4_nat_1), [interesting(0.9),axiom,file(nat_1,k4_nat_1)]). fof(t46_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ~ ( ~ r1_xreal_0(A,0) & r1_xreal_0(A,k4_nat_1(B,A)) ) ) ) ), file(nat_1,t46_nat_1), [interesting(0.9),axiom,file(nat_1,t46_nat_1)]). fof(e4_5_1__ami_5,plain,( ~ r1_xreal_0(2,k4_nat_1(c2_5_1__ami_5,2)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_5_1__ami_5,e1_5_1__ami_5])],[reflexivity_r1_tarski,cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc1_arytm_3,cc2_arytm_3,cc3_arytm_3,fc1_ordinal2,fc1_xboole_0,fc2_finseq_1,rc1_arytm_3,rc1_int_1,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_c1_5_1__ami_5,cc1_funct_1,cc2_int_1,cc4_int_1,fc1_numbers,rc1_xboole_0,rc2_int_1,rc2_xboole_0,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,dt_k4_nat_1,dt_c2_5_1__ami_5,de_c2_5_1__ami_5,cc3_int_1,spc0_numerals,spc2_numerals,spc0_boole,spc2_boole,t46_nat_1,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r0_r0]), [interesting(0.65),file(ami_5,e4_5_1__ami_5),[file(ami_5,e4_5_1__ami_5)]]). fof(t71_nat_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( ~ r1_xreal_0(2,A) & A != 0 & A != 1 ) ) ), file(nat_1,t71_nat_1), [interesting(0.9),axiom,file(nat_1,t71_nat_1)]). fof(e5_5_1__ami_5,plain, ( k4_nat_1(c2_5_1__ami_5,2) = 0 | k4_nat_1(c2_5_1__ami_5,2) = 1 ), inference(mizar_by,[status(thm),assumptions([dt_c1_5_1__ami_5,e1_5_1__ami_5])],[cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,cc1_arytm_3,cc2_arytm_3,cc4_int_1,fc1_xboole_0,fc2_finseq_1,rc1_arytm_3,rc1_int_1,rc2_int_1,t1_subset,t2_real,t3_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,dt_c1_5_1__ami_5,cc1_funct_1,cc2_int_1,cc3_arytm_3,cc3_int_1,fc1_ordinal2,rc1_xboole_0,rc2_xboole_0,t1_numerals,t1_real,t2_subset,t3_subset,t4_real,t6_boole,t7_boole,t8_boole,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k4_nat_1,dt_k5_numbers,dt_m2_subset_1,dt_c2_5_1__ami_5,de_c2_5_1__ami_5,fc1_numbers,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r2,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e4_5_1__ami_5,t71_nat_1,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1]), [interesting(0.65),file(ami_5,e5_5_1__ami_5),[file(ami_5,e5_5_1__ami_5)]]). fof(t47_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( ~ r1_xreal_0(A,0) => B = k2_xcmplx_0(k3_xcmplx_0(A,k3_nat_1(B,A)),k4_nat_1(B,A)) ) ) ) ), file(nat_1,t47_nat_1), [interesting(0.9),axiom,file(nat_1,t47_nat_1)]). fof(e6_5_1__ami_5,plain, ( c2_5_1__ami_5 = k1_nat_1(k2_nat_1(2,k3_nat_1(c2_5_1__ami_5,2)),0) | c2_5_1__ami_5 = k1_nat_1(k2_nat_1(2,k3_nat_1(c2_5_1__ami_5,2)),1) ), inference(mizar_by,[status(thm),assumptions([dt_c1_5_1__ami_5,e1_5_1__ami_5])],[reflexivity_r1_tarski,cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc1_arytm_3,cc2_arytm_3,cc3_arytm_3,fc1_ordinal2,fc1_xboole_0,fc2_finseq_1,rc1_arytm_3,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_c1_5_1__ami_5,cc1_funct_1,cc2_int_1,cc4_int_1,fc1_int_1,fc1_numbers,fc2_int_1,fc6_int_1,fc7_int_1,rc1_int_1,rc1_xboole_0,rc2_int_1,rc2_xboole_0,spc5_arithm,spc6_arithm,spc7_arithm,t1_arithm,t1_numerals,t1_real,t2_arithm,t2_subset,t3_arithm,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k2_nat_1,commutativity_k2_xcmplx_0,commutativity_k3_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k2_nat_1,dt_k1_nat_1,dt_k2_nat_1,dt_k2_xcmplx_0,dt_k3_nat_1,dt_k3_xcmplx_0,dt_k4_nat_1,dt_c2_5_1__ami_5,de_c2_5_1__ami_5,cc3_int_1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r2,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_r2_r2,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqRealAdd__k2_xcmplx_0__r2_r0_r2,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e5_5_1__ami_5,t47_nat_1,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r1_r0]), [interesting(0.65),file(ami_5,e6_5_1__ami_5),[file(ami_5,e6_5_1__ami_5)]]). fof(e1_5_1_1__ami_5,plain,( ~ ( c1_5_1__ami_5 != 0 & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => c1_5_1__ami_5 != k1_nat_1(k2_nat_1(2,A),1) ) & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( c1_5_1__ami_5 = k2_nat_1(2,A) & ~ r1_xreal_0(A,0) ) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_5_1__ami_5,e1_5_1__ami_5])],[cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,cc1_arytm_3,cc2_arytm_3,cc4_int_1,fc1_int_1,fc1_xboole_0,fc2_finseq_1,fc2_int_1,fc3_int_1,fc4_int_1,fc6_int_1,fc7_int_1,fc8_int_1,rc1_arytm_3,rc1_int_1,rc2_int_1,t1_subset,t2_real,t3_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc1_funct_1,cc2_int_1,cc3_arytm_3,cc3_int_1,fc1_ordinal2,fc5_int_1,fc9_int_1,rc1_xboole_0,rc2_xboole_0,spc1_arithm,spc2_arithm,spc5_arithm,spc6_arithm,spc7_arithm,spc8_arithm,spc9_arithm,t1_arithm,t1_numerals,t1_real,t2_arithm,t2_subset,t3_arithm,t3_subset,t4_arithm,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k2_nat_1,commutativity_k2_xcmplx_0,commutativity_k3_xcmplx_0,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k2_nat_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_nat_1,dt_k1_numbers,dt_k2_nat_1,dt_k2_xcmplx_0,dt_k3_nat_1,dt_k3_xcmplx_0,dt_k4_xcmplx_0,dt_k5_numbers,dt_k6_xcmplx_0,dt_m2_subset_1,dt_c1_5_1__ami_5,dt_c2_5_1__ami_5,de_c2_5_1__ami_5,fc1_numbers,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r0_rm2,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r1_rm2,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r2_rm1,rqLessOrEqual__r1_xreal_0__r2_rm2,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_r2,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqLessOrEqual__r1_xreal_0__rm1_rm2,rqLessOrEqual__r1_xreal_0__rm2_r0,rqLessOrEqual__r1_xreal_0__rm2_r1,rqLessOrEqual__r1_xreal_0__rm2_r2,rqLessOrEqual__r1_xreal_0__rm2_rm1,rqLessOrEqual__r1_xreal_0__rm2_rm2,rqRealAdd__k2_xcmplx_0__r0_r2_r2,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,rqRealAdd__k2_xcmplx_0__r0_rm2_rm2,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__r1_rm2_rm1,rqRealAdd__k2_xcmplx_0__r2_r0_r2,rqRealAdd__k2_xcmplx_0__r2_rm1_r1,rqRealAdd__k2_xcmplx_0__r2_rm2_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0,rqRealAdd__k2_xcmplx_0__rm1_r2_r1,rqRealAdd__k2_xcmplx_0__rm1_rm1_rm2,rqRealAdd__k2_xcmplx_0__rm2_r0_rm2,rqRealAdd__k2_xcmplx_0__rm2_r1_rm1,rqRealAdd__k2_xcmplx_0__rm2_r2_r0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_rm2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_rm2_rm2,rqRealMult__k3_xcmplx_0__r2_r1_r2,rqRealMult__k3_xcmplx_0__rm2_r0_r0,rqRealMult__k3_xcmplx_0__rm2_r1_rm2,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e3_5_1__ami_5,e6_5_1__ami_5,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealNeg__k4_xcmplx_0__rm2_r2,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r1_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1]), [interesting(0.5),file(ami_5,e1_5_1_1__ami_5),[file(ami_5,e1_5_1_1__ami_5)]]). fof(i2_5_1__ami_5,plain,( r2_hidden(c1_5_1__ami_5,k2_xboole_0(k2_xboole_0(k6_domain_1(k5_numbers,0),a_0_0_ami_5),a_0_1_ami_5)) ), inference(percases,[status(thm),assumptions([dt_c1_5_1__ami_5,e1_5_1__ami_5])],[i1_5_1_1__ami_5,i2_5_1_1__ami_5,i3_5_1_1__ami_5,e1_5_1_1__ami_5]), [interesting(0.65),file(ami_5,i2_5_1__ami_5),[file(ami_5,i2_5_1__ami_5)]]). fof(i1_5_1__ami_5,plain,( ~ ( r2_hidden(c1_5_1__ami_5,k5_numbers) & ~ r2_hidden(c1_5_1__ami_5,k2_xboole_0(k2_xboole_0(k6_domain_1(k5_numbers,0),a_0_0_ami_5),a_0_1_ami_5)) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_5_1__ami_5]),discharge_asm(discharge,[e1_5_1__ami_5])],[e1_5_1__ami_5,i2_5_1__ami_5]), [interesting(0.65),file(ami_5,i1_5_1__ami_5),[file(ami_5,i1_5_1__ami_5)]]). fof(i1_5_1_tmp__ami_5,plain,( ~ ( r2_hidden(c1_5_1__ami_5,k5_numbers) & ~ r2_hidden(c1_5_1__ami_5,k2_xboole_0(k2_xboole_0(k6_domain_1(k5_numbers,0),a_0_0_ami_5),a_0_1_ami_5)) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_5_1__ami_5])],[dt_c1_5_1__ami_5,i1_5_1__ami_5]), [interesting(0.8),e1_5__ami_5]). fof(e1_5__ami_5,plain,( r1_tarski(k5_numbers,k2_xboole_0(k2_xboole_0(k6_domain_1(k5_numbers,0),a_0_0_ami_5),a_0_1_ami_5)) ), inference(let,[status(thm),assumptions([])],[i1_5_1_tmp__ami_5,commutativity_k2_xcmplx_0,commutativity_k3_xcmplx_0,dt_k2_xcmplx_0,dt_k3_xcmplx_0,cc1_arytm_3,cc1_setfam_1,cc2_arytm_3,cc3_int_1,cc4_int_1,fc1_int_1,fc2_int_1,fc4_setfam_1,fc6_int_1,fc7_int_1,rc1_arytm_3,rc1_int_1,rc1_setfam_1,rc2_int_1,commutativity_k1_nat_1,commutativity_k2_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k2_nat_1,redefinition_m2_subset_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_tarski,dt_k1_zfmisc_1,dt_k2_nat_1,dt_k5_ordinal2,dt_m1_subset_1,dt_m2_subset_1,cc1_funct_1,cc2_int_1,cc3_arytm_3,fc1_numbers,fc1_ordinal2,fc2_setfam_1,fc2_xboole_0,fc3_xboole_0,rc1_xboole_0,rc2_xboole_0,spc1_numerals,spc2_numerals,spc1_boole,spc2_boole,commutativity_k2_xboole_0,idempotence_k2_xboole_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,redefinition_k5_numbers,redefinition_k6_domain_1,dt_k2_xboole_0,dt_k5_numbers,dt_k6_domain_1,spc0_numerals,spc0_boole,t2_tarski,fraenkel_a_0_0_ami_5,fraenkel_a_0_1_ami_5,d3_tarski,dh_c1_5_1__ami_5]), [interesting(0.8),file(ami_5,e1_5__ami_5),[file(ami_5,e1_5__ami_5)]]). fof(d2_ami_2,definition,( k2_ami_2 = a_0_0_ami_2 ), file(ami_2,d2_ami_2), [interesting(0.9),axiom,file(ami_2,d2_ami_2)]). fof(d3_ami_2,definition,( k3_ami_2 = a_0_1_ami_2 ), file(ami_2,d3_ami_2), [interesting(0.9),axiom,file(ami_2,d3_ami_2)]). fof(d10_xboole_0,definition,( ! [A,B] : ( A = B <=> ( r1_tarski(A,B) & r1_tarski(B,A) ) ) ), file(xboole_0,d10_xboole_0), [interesting(0.9),axiom,file(xboole_0,d10_xboole_0)]). fof(e5_5__ami_5,plain,( u1_struct_0(k1_ami_3) = k2_xboole_0(k2_xboole_0(k1_struct_0(k1_ami_3,k2_ami_1(k1_tarski(k4_numbers),k1_ami_3)),k2_ami_2),k3_ami_2) ), inference(mizar_by,[status(thm),assumptions([])],[cc1_finseq_1,fc2_fraenkel,rc1_finseq_1,rc2_finseq_1,rc3_finseq_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,commutativity_k2_xcmplx_0,commutativity_k3_xcmplx_0,existence_m1_fraenkel,dt_k1_xboole_0,dt_k2_xcmplx_0,dt_k3_xcmplx_0,dt_m1_fraenkel,dt_u2_ami_1,dt_u3_ami_1,dt_u4_ami_1,dt_u5_ami_1,dt_u6_ami_1,cc1_arytm_3,cc1_fraenkel,cc1_setfam_1,cc2_arytm_3,cc4_int_1,fc1_int_1,fc1_xboole_0,fc2_ami_1,fc2_finseq_1,fc2_int_1,fc4_ami_1,fc4_setfam_1,fc6_int_1,fc7_int_1,rc1_arytm_3,rc1_fraenkel,rc1_int_1,rc1_setfam_1,rc2_int_1,rc3_ami_1,rc3_funct_1,rc7_ami_1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_r2_r2,rqRealAdd__k2_xcmplx_0__r0_r9_r9,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqRealAdd__k2_xcmplx_0__r2_r0_r2,rqRealAdd__k2_xcmplx_0__r9_r0_r9,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r1_r9_r9,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,rqRealMult__k3_xcmplx_0__r9_r1_r9,spc5_arithm,spc6_arithm,spc7_arithm,t1_arithm,t1_boole,t1_real,t2_arithm,t2_real,t3_arithm,t3_real,t4_real,t5_real,t6_real,t7_real,t8_real,commutativity_k1_nat_1,commutativity_k2_nat_1,commutativity_k2_tarski,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,abstractness_v1_ami_1,existence_l1_ami_1,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k1_fraenkel,redefinition_k1_nat_1,redefinition_k2_nat_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k13_finseq_1,dt_k1_fraenkel,dt_k1_funct_2,dt_k1_nat_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k2_nat_1,dt_k2_tarski,dt_k2_zfmisc_1,dt_k3_tarski,dt_k4_card_3,dt_k5_ordinal2,dt_l1_ami_1,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_ami_1,cc1_funct_1,cc1_int_1,cc1_relset_1,cc2_funct_1,cc2_int_1,cc3_arytm_3,cc3_int_1,fc16_finseq_1,fc1_fraenkel,fc1_numbers,fc1_ordinal2,fc1_struct_0,fc2_setfam_1,fc2_xboole_0,fc3_ami_1,fc3_setfam_1,fc3_xboole_0,fc9_finseq_1,rc1_ami_1,rc1_funct_1,rc1_xboole_0,rc2_ami_1,rc2_funct_1,rc2_xboole_0,rc3_struct_0,rc5_ami_1,rc5_struct_0,rc6_ami_1,rc8_ami_1,rc9_ami_1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r0_r9,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_r9,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r2_r9,rqLessOrEqual__r1_xreal_0__r9_r0,rqLessOrEqual__r1_xreal_0__r9_r1,rqLessOrEqual__r1_xreal_0__r9_r2,rqLessOrEqual__r1_xreal_0__r9_r9,spc1_boole,spc1_numerals,spc2_boole,spc2_numerals,t1_numerals,t1_subset,t2_subset,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,spc1_numerals,spc2_numerals,spc1_boole,spc2_boole,free_g1_ami_1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,reflexivity_r1_tarski,redefinition_k1_struct_0,redefinition_k5_numbers,redefinition_k6_domain_1,dt_g1_ami_1,dt_k17_ami_2,dt_k1_ami_3,dt_k1_gr_cy_1,dt_k1_struct_0,dt_k1_tarski,dt_k2_ami_1,dt_k2_ami_2,dt_k2_xboole_0,dt_k3_ami_2,dt_k4_ami_2,dt_k4_numbers,dt_k5_ami_2,dt_k5_numbers,dt_k6_domain_1,dt_u1_struct_0,fc1_ami_2,fc1_ami_3,fc2_ami_2,fc2_ami_3,fc3_ami_2,fc3_ami_3,fc4_numbers,fc5_ami_3,t3_subset,t2_tarski,fraenkel_a_0_0_ami_2,fraenkel_a_0_0_ami_5,fraenkel_a_0_1_ami_2,fraenkel_a_0_1_ami_5,d5_ami_1,spc0_numerals,spc9_numerals,spc0_boole,spc9_boole,e4_5__ami_5,e1_5__ami_5,d2_ami_2,d3_ami_2,d1_ami_3,d10_xboole_0]), [interesting(0.8),file(ami_5,e5_5__ami_5),[file(ami_5,e5_5__ami_5)]]). fof(i1_5__ami_5,theorem,( $true ), introduced(tautology,[file(ami_5,i1_5__ami_5)]), [interesting(0.8),trivial,file(ami_5,i1_5__ami_5)]). fof(t23_ami_5,theorem,( u1_struct_0(k1_ami_3) = k2_xboole_0(k2_xboole_0(k1_struct_0(k1_ami_3,k2_ami_1(k1_tarski(k4_numbers),k1_ami_3)),k2_ami_2),k3_ami_2) ), inference(conclusion,[status(thm),assumptions([])],[rc3_funct_1,cc1_fraenkel,cc2_funct_1,fc3_ami_1,rc1_fraenkel,rc1_funct_1,rc2_funct_1,commutativity_k2_tarski,redefinition_m2_relset_1,dt_k13_finseq_1,dt_k1_funct_2,dt_k2_tarski,dt_k2_zfmisc_1,dt_k3_tarski,dt_k4_card_3,dt_m1_relset_1,dt_m2_relset_1,cc1_relset_1,fc16_finseq_1,fc1_fraenkel,fc3_setfam_1,fc4_ami_1,fc9_finseq_1,free_g1_ami_1,dt_g1_ami_1,dt_k1_numbers,dt_k5_ordinal2,dt_u1_ami_1,dt_u2_ami_1,dt_u3_ami_1,dt_u4_ami_1,dt_u5_ami_1,dt_u6_ami_1,cc1_arytm_3,cc1_setfam_1,cc2_arytm_3,cc3_arytm_3,cc3_int_1,cc4_int_1,fc1_numbers,fc1_ordinal2,fc2_ami_1,fc4_setfam_1,rc1_arytm_3,rc1_int_1,rc1_setfam_1,rc2_int_1,rc3_ami_1,rc7_ami_1,abstractness_v1_ami_1,redefinition_k5_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_l1_ami_1,dt_l1_struct_0,dt_m1_subset_1,cc1_funct_1,cc1_int_1,cc2_int_1,fc1_struct_0,fc2_setfam_1,fc2_xboole_0,fc3_xboole_0,rc1_ami_1,rc1_xboole_0,rc2_ami_1,rc2_xboole_0,rc3_struct_0,rc5_ami_1,rc5_struct_0,rc6_ami_1,rc8_ami_1,rc9_ami_1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,redefinition_k1_struct_0,dt_k1_ami_3,dt_k1_struct_0,dt_k1_tarski,dt_k2_ami_1,dt_k2_ami_2,dt_k2_xboole_0,dt_k3_ami_2,dt_k4_numbers,dt_u1_struct_0,fc1_ami_2,fc1_ami_3,fc2_ami_2,fc2_ami_3,fc3_ami_3,fc4_numbers,fc5_ami_3,e5_5__ami_5,i1_5__ami_5]), [interesting(1),file(ami_5,t23_ami_5),[file(ami_5,t23_ami_5)]]).