% Mizar ND problem: t7_bhsp_3,bhsp_3,230,38 fof(dh_c1_9__bhsp_3,definition, ( ( ( ~ v3_struct_0(c1_9__bhsp_3) & v3_rlvect_1(c1_9__bhsp_3) & v4_rlvect_1(c1_9__bhsp_3) & v5_rlvect_1(c1_9__bhsp_3) & v6_rlvect_1(c1_9__bhsp_3) & v7_rlvect_1(c1_9__bhsp_3) & v2_bhsp_1(c1_9__bhsp_3) & l1_bhsp_1(c1_9__bhsp_3) ) => ! [A] : ( m1_subset_1(A,u1_struct_0(c1_9__bhsp_3)) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(c1_9__bhsp_3)) & m2_relset_1(B,k5_numbers,u1_struct_0(c1_9__bhsp_3)) ) => ( v1_bhsp_3(B,c1_9__bhsp_3) => v1_bhsp_3(k7_bhsp_1(c1_9__bhsp_3,B,A),c1_9__bhsp_3) ) ) ) ) => ! [C] : ( ( ~ v3_struct_0(C) & v3_rlvect_1(C) & v4_rlvect_1(C) & v5_rlvect_1(C) & v6_rlvect_1(C) & v7_rlvect_1(C) & v2_bhsp_1(C) & l1_bhsp_1(C) ) => ! [D] : ( m1_subset_1(D,u1_struct_0(C)) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,k5_numbers,u1_struct_0(C)) & m2_relset_1(E,k5_numbers,u1_struct_0(C)) ) => ( v1_bhsp_3(E,C) => v1_bhsp_3(k7_bhsp_1(C,E,D),C) ) ) ) ) ), introduced(definition,[new_symbol(c1_9__bhsp_3),file(bhsp_3,c1_9__bhsp_3)]), [interesting(0.8),axiom,file(bhsp_3,c1_9__bhsp_3)]). fof(dh_c2_9__bhsp_3,definition, ( ( m1_subset_1(c2_9__bhsp_3,u1_struct_0(c1_9__bhsp_3)) => ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,u1_struct_0(c1_9__bhsp_3)) & m2_relset_1(A,k5_numbers,u1_struct_0(c1_9__bhsp_3)) ) => ( v1_bhsp_3(A,c1_9__bhsp_3) => v1_bhsp_3(k7_bhsp_1(c1_9__bhsp_3,A,c2_9__bhsp_3),c1_9__bhsp_3) ) ) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(c1_9__bhsp_3)) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,u1_struct_0(c1_9__bhsp_3)) & m2_relset_1(C,k5_numbers,u1_struct_0(c1_9__bhsp_3)) ) => ( v1_bhsp_3(C,c1_9__bhsp_3) => v1_bhsp_3(k7_bhsp_1(c1_9__bhsp_3,C,B),c1_9__bhsp_3) ) ) ) ), introduced(definition,[new_symbol(c2_9__bhsp_3),file(bhsp_3,c2_9__bhsp_3)]), [interesting(0.8),axiom,file(bhsp_3,c2_9__bhsp_3)]). fof(dh_c3_9__bhsp_3,definition, ( ( ( v1_funct_1(c3_9__bhsp_3) & v1_funct_2(c3_9__bhsp_3,k5_numbers,u1_struct_0(c1_9__bhsp_3)) & m2_relset_1(c3_9__bhsp_3,k5_numbers,u1_struct_0(c1_9__bhsp_3)) ) => ( v1_bhsp_3(c3_9__bhsp_3,c1_9__bhsp_3) => v1_bhsp_3(k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),c1_9__bhsp_3) ) ) => ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,u1_struct_0(c1_9__bhsp_3)) & m2_relset_1(A,k5_numbers,u1_struct_0(c1_9__bhsp_3)) ) => ( v1_bhsp_3(A,c1_9__bhsp_3) => v1_bhsp_3(k7_bhsp_1(c1_9__bhsp_3,A,c2_9__bhsp_3),c1_9__bhsp_3) ) ) ), introduced(definition,[new_symbol(c3_9__bhsp_3),file(bhsp_3,c3_9__bhsp_3)]), [interesting(0.8),axiom,file(bhsp_3,c3_9__bhsp_3)]). fof(e1_9__bhsp_3,assumption,( v1_bhsp_3(c3_9__bhsp_3,c1_9__bhsp_3) ), introduced(assumption,[file(bhsp_3,e1_9__bhsp_3)]), [interesting(0.8),axiom,file(bhsp_3,e1_9__bhsp_3)]). fof(dt_l2_struct_0,axiom,( ! [A] : ( l2_struct_0(A) => l1_struct_0(A) ) ), file(struct_0,l2_struct_0), [interesting(0.9),axiom,file(struct_0,l2_struct_0)]). 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(rc4_struct_0,theorem,( ? [A] : ( l2_struct_0(A) & ~ v3_struct_0(A) ) ), file(struct_0,rc4_struct_0), [interesting(0.9),axiom,file(struct_0,rc4_struct_0)]). fof(dt_l1_rlvect_1,axiom,( ! [A] : ( l1_rlvect_1(A) => l2_struct_0(A) ) ), file(rlvect_1,l1_rlvect_1), [interesting(0.9),axiom,file(rlvect_1,l1_rlvect_1)]). fof(cc12_membered,theorem,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_rat_1(B) ) ) ) ), file(membered,cc12_membered), [interesting(0.9),axiom,file(membered,cc12_membered)]). fof(cc13_membered,theorem,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc13_membered), [interesting(0.9),axiom,file(membered,cc13_membered)]). fof(cc14_membered,theorem,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v4_ordinal2(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc14_membered), [interesting(0.9),axiom,file(membered,cc14_membered)]). fof(cc18_membered,theorem,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) ) ) ) ), file(membered,cc18_membered), [interesting(0.9),axiom,file(membered,cc18_membered)]). fof(cc19_membered,theorem,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) ) ) ) ), file(membered,cc19_membered), [interesting(0.9),axiom,file(membered,cc19_membered)]). 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_membered,theorem,( ! [A] : ( v5_membered(A) => v4_membered(A) ) ), file(membered,cc1_membered), [interesting(0.9),axiom,file(membered,cc1_membered)]). fof(cc1_xreal_0,theorem,( ! [A] : ( v4_ordinal2(A) => v1_xreal_0(A) ) ), file(xreal_0,cc1_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc1_xreal_0)]). fof(cc20_membered,theorem,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) & v5_membered(B) ) ) ) ), file(membered,cc20_membered), [interesting(0.9),axiom,file(membered,cc20_membered)]). 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(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(cc2_membered,theorem,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), [interesting(0.9),axiom,file(membered,cc2_membered)]). fof(cc3_membered,theorem,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), [interesting(0.9),axiom,file(membered,cc3_membered)]). fof(cc4_xreal_0,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v2_xreal_0(A) ) ) ), file(xreal_0,cc4_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc4_xreal_0)]). fof(cc5_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & v3_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) ) ) ), file(xreal_0,cc5_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc5_xreal_0)]). fof(cc8_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(xreal_0,cc8_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc8_xreal_0)]). 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_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(rc1_membered,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ), file(membered,rc1_membered), [interesting(0.9),axiom,file(membered,rc1_membered)]). fof(rc1_xreal_0,theorem,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(xreal_0,rc1_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc1_xreal_0)]). 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(rc2_xreal_0,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(xreal_0,rc2_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc2_xreal_0)]). fof(rc3_xreal_0,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & v3_xreal_0(A) ) ), file(xreal_0,rc3_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc3_xreal_0)]). fof(rc4_xreal_0,theorem,( ? [A] : ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(xreal_0,rc4_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc4_xreal_0)]). fof(dt_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), [interesting(0.9),axiom,file(funct_1,k1_funct_1)]). 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_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,k2_zfmisc_1)]). fof(dt_k4_bhsp_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_rlvect_1(A) & v2_bhsp_1(A) & l1_bhsp_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k4_bhsp_1(A,B,C),k1_numbers) ) ), file(bhsp_1,k4_bhsp_1), [interesting(0.9),axiom,file(bhsp_1,k4_bhsp_1)]). fof(dt_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), [interesting(0.9),axiom,file(ordinal2,k5_ordinal2)]). 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_l2_rlvect_1,axiom,( ! [A] : ( l2_rlvect_1(A) => l1_rlvect_1(A) ) ), file(rlvect_1,l2_rlvect_1), [interesting(0.9),axiom,file(rlvect_1,l2_rlvect_1)]). fof(dt_m1_relset_1,axiom,( $true ), file(relset_1,m1_relset_1), [interesting(0.9),axiom,file(relset_1,m1_relset_1)]). fof(cc10_membered,theorem,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,A) => v1_xcmplx_0(B) ) ) ), file(membered,cc10_membered), [interesting(0.9),axiom,file(membered,cc10_membered)]). fof(cc11_membered,theorem,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) ) ) ) ), file(membered,cc11_membered), [interesting(0.9),axiom,file(membered,cc11_membered)]). fof(cc15_membered,theorem,( ! [A] : ( v1_xboole_0(A) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc15_membered), [interesting(0.9),axiom,file(membered,cc15_membered)]). fof(cc16_membered,theorem,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_membered(B) ) ) ), file(membered,cc16_membered), [interesting(0.9),axiom,file(membered,cc16_membered)]). fof(cc17_membered,theorem,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) ) ) ) ), file(membered,cc17_membered), [interesting(0.9),axiom,file(membered,cc17_membered)]). 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_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(cc2_xreal_0,theorem,( ! [A] : ( v1_xreal_0(A) => v1_xcmplx_0(A) ) ), file(xreal_0,cc2_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc2_xreal_0)]). 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_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & v2_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc3_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc3_xreal_0)]). fof(cc4_membered,theorem,( ! [A] : ( v2_membered(A) => v1_membered(A) ) ), file(membered,cc4_membered), [interesting(0.9),axiom,file(membered,cc4_membered)]). fof(cc6_membered,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( v1_membered(A) & v2_membered(A) ) ) ), file(membered,cc6_membered), [interesting(0.9),axiom,file(membered,cc6_membered)]). fof(cc6_xreal_0,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v3_xreal_0(A) ) ) ), file(xreal_0,cc6_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc6_xreal_0)]). fof(cc7_xreal_0,theorem,( ! [A] : ( ( v1_xboole_0(A) & v1_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc7_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc7_xreal_0)]). fof(cc9_membered,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k5_numbers)) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc9_membered), [interesting(0.9),axiom,file(membered,cc9_membered)]). 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(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(fc5_membered,theorem, ( v1_membered(k5_ordinal2) & v2_membered(k5_ordinal2) & v3_membered(k5_ordinal2) & v4_membered(k5_ordinal2) & v5_membered(k5_ordinal2) ), file(membered,fc5_membered), [interesting(0.9),axiom,file(membered,fc5_membered)]). 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_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(commutativity_k5_bhsp_1,theorem,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_rlvect_1(A) & v2_bhsp_1(A) & l1_bhsp_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => k5_bhsp_1(A,B,C) = k5_bhsp_1(A,C,B) ) ), file(bhsp_1,k5_bhsp_1), [interesting(0.9),axiom,file(bhsp_1,k5_bhsp_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(redefinition_k2_normsp_1,definition,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(A)) & m1_relset_1(B,k5_numbers,u1_struct_0(A)) & m1_subset_1(C,k5_numbers) ) => k2_normsp_1(A,B,C) = k1_funct_1(B,C) ) ), file(normsp_1,k2_normsp_1), [interesting(0.9),axiom,file(normsp_1,k2_normsp_1)]). fof(redefinition_k5_bhsp_1,definition,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_rlvect_1(A) & v2_bhsp_1(A) & l1_bhsp_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => k5_bhsp_1(A,B,C) = k4_bhsp_1(A,B,C) ) ), file(bhsp_1,k5_bhsp_1), [interesting(0.9),axiom,file(bhsp_1,k5_bhsp_1)]). fof(redefinition_k5_numbers,definition,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), [interesting(0.9),axiom,file(numbers,k5_numbers)]). 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(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_numbers,axiom,( $true ), file(numbers,k1_numbers), [interesting(0.9),axiom,file(numbers,k1_numbers)]). fof(dt_k2_normsp_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(A)) & m1_relset_1(B,k5_numbers,u1_struct_0(A)) & m1_subset_1(C,k5_numbers) ) => m1_subset_1(k2_normsp_1(A,B,C),u1_struct_0(A)) ) ), file(normsp_1,k2_normsp_1), [interesting(0.9),axiom,file(normsp_1,k2_normsp_1)]). fof(dt_k5_bhsp_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_rlvect_1(A) & v2_bhsp_1(A) & l1_bhsp_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k5_bhsp_1(A,B,C),k1_numbers) ) ), file(bhsp_1,k5_bhsp_1), [interesting(0.9),axiom,file(bhsp_1,k5_bhsp_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_k7_bhsp_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_rlvect_1(A) & v2_bhsp_1(A) & l1_bhsp_1(A) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(A)) & m1_relset_1(B,k5_numbers,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => ( v1_funct_1(k7_bhsp_1(A,B,C)) & v1_funct_2(k7_bhsp_1(A,B,C),k5_numbers,u1_struct_0(A)) & m2_relset_1(k7_bhsp_1(A,B,C),k5_numbers,u1_struct_0(A)) ) ) ), file(bhsp_1,k7_bhsp_1), [interesting(0.9),axiom,file(bhsp_1,k7_bhsp_1)]). fof(dt_l1_bhsp_1,axiom,( ! [A] : ( l1_bhsp_1(A) => l2_rlvect_1(A) ) ), file(bhsp_1,l1_bhsp_1), [interesting(0.9),axiom,file(bhsp_1,l1_bhsp_1)]). fof(dt_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), [interesting(0.9),axiom,file(subset_1,m1_subset_1)]). fof(dt_m2_relset_1,axiom,( ! [A,B,C] : ( m2_relset_1(C,A,B) => m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ), file(relset_1,m2_relset_1), [interesting(0.9),axiom,file(relset_1,m2_relset_1)]). fof(dt_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m2_subset_1(C,A,B) => m1_subset_1(C,A) ) ) ), file(subset_1,m2_subset_1), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(dt_u1_struct_0,axiom,( $true ), file(struct_0,u1_struct_0), [interesting(0.9),axiom,file(struct_0,u1_struct_0)]). fof(dt_c1_9__bhsp_3,assumption, ( ~ v3_struct_0(c1_9__bhsp_3) & v3_rlvect_1(c1_9__bhsp_3) & v4_rlvect_1(c1_9__bhsp_3) & v5_rlvect_1(c1_9__bhsp_3) & v6_rlvect_1(c1_9__bhsp_3) & v7_rlvect_1(c1_9__bhsp_3) & v2_bhsp_1(c1_9__bhsp_3) & l1_bhsp_1(c1_9__bhsp_3) ), introduced(assumption,[file(bhsp_3,c1_9__bhsp_3)]), [interesting(0.8),axiom,file(bhsp_3,c1_9__bhsp_3)]). fof(dt_c2_9__bhsp_3,assumption,( m1_subset_1(c2_9__bhsp_3,u1_struct_0(c1_9__bhsp_3)) ), introduced(assumption,[file(bhsp_3,c2_9__bhsp_3)]), [interesting(0.8),axiom,file(bhsp_3,c2_9__bhsp_3)]). fof(dt_c3_9__bhsp_3,assumption, ( v1_funct_1(c3_9__bhsp_3) & v1_funct_2(c3_9__bhsp_3,k5_numbers,u1_struct_0(c1_9__bhsp_3)) & m2_relset_1(c3_9__bhsp_3,k5_numbers,u1_struct_0(c1_9__bhsp_3)) ), introduced(assumption,[file(bhsp_3,c3_9__bhsp_3)]), [interesting(0.8),axiom,file(bhsp_3,c3_9__bhsp_3)]). fof(dt_c4_9__bhsp_3,assumption,( m1_subset_1(c4_9__bhsp_3,k1_numbers) ), introduced(assumption,[file(bhsp_3,c4_9__bhsp_3)]), [interesting(0.8),axiom,file(bhsp_3,c4_9__bhsp_3)]). fof(fc2_membered,theorem, ( ~ v1_xboole_0(k1_numbers) & v1_membered(k1_numbers) & v2_membered(k1_numbers) ), file(membered,fc2_membered), [interesting(0.9),axiom,file(membered,fc2_membered)]). 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(spc0_boole,theorem,( v1_xboole_0(0) ), file(boole,spc0_boole), [interesting(0.9),axiom,file(boole,spc0_boole)]). fof(d1_bhsp_3,definition,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_rlvect_1(A) & v2_bhsp_1(A) & l1_bhsp_1(A) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(A)) & m2_relset_1(B,k5_numbers,u1_struct_0(A)) ) => ( v1_bhsp_3(B,A) <=> ! [C] : ( m1_subset_1(C,k1_numbers) => ~ ( ~ r1_xreal_0(C,0) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ? [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) & ? [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) & r1_xreal_0(D,E) & r1_xreal_0(D,F) & r1_xreal_0(C,k5_bhsp_1(A,k2_normsp_1(A,B,E),k2_normsp_1(A,B,F))) ) ) ) ) ) ) ) ) ), file(bhsp_3,d1_bhsp_3), [interesting(0.9),axiom,file(bhsp_3,d1_bhsp_3)]). fof(dh_c4_9__bhsp_3,definition, ( ( m1_subset_1(c4_9__bhsp_3,k1_numbers) => ~ ( ~ r1_xreal_0(c4_9__bhsp_3,0) & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ? [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) & ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & r1_xreal_0(A,B) & r1_xreal_0(A,C) & r1_xreal_0(c4_9__bhsp_3,k5_bhsp_1(c1_9__bhsp_3,k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),B),k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),C))) ) ) ) ) ) => ! [D] : ( m1_subset_1(D,k1_numbers) => ~ ( ~ r1_xreal_0(D,0) & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ? [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) & ? [G] : ( m2_subset_1(G,k1_numbers,k5_numbers) & r1_xreal_0(E,F) & r1_xreal_0(E,G) & r1_xreal_0(D,k5_bhsp_1(c1_9__bhsp_3,k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),F),k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),G))) ) ) ) ) ) ), introduced(definition,[new_symbol(c4_9__bhsp_3),file(bhsp_3,c4_9__bhsp_3)]), [interesting(0.8),axiom,file(bhsp_3,c4_9__bhsp_3)]). fof(e2_9__bhsp_3,assumption,( ~ r1_xreal_0(c4_9__bhsp_3,0) ), introduced(assumption,[file(bhsp_3,e2_9__bhsp_3)]), [interesting(0.8),axiom,file(bhsp_3,e2_9__bhsp_3)]). fof(reflexivity_r1_tarski,theorem,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), [interesting(0.9),axiom,file(tarski,r1_tarski)]). 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(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]). fof(fc6_membered,theorem, ( v1_xboole_0(k1_xboole_0) & v1_membered(k1_xboole_0) & v2_membered(k1_xboole_0) & v3_membered(k1_xboole_0) & v4_membered(k1_xboole_0) & v5_membered(k1_xboole_0) ), file(membered,fc6_membered), [interesting(0.9),axiom,file(membered,fc6_membered)]). 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(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(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(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(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(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(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(dh_c5_9__bhsp_3,definition, ( ? [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(A,B) & r1_xreal_0(A,C) & r1_xreal_0(c4_9__bhsp_3,k5_bhsp_1(c1_9__bhsp_3,k2_normsp_1(c1_9__bhsp_3,c3_9__bhsp_3,B),k2_normsp_1(c1_9__bhsp_3,c3_9__bhsp_3,C))) ) ) ) ) => ( m2_subset_1(c5_9__bhsp_3,k1_numbers,k5_numbers) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c5_9__bhsp_3,D) & r1_xreal_0(c5_9__bhsp_3,E) & r1_xreal_0(c4_9__bhsp_3,k5_bhsp_1(c1_9__bhsp_3,k2_normsp_1(c1_9__bhsp_3,c3_9__bhsp_3,D),k2_normsp_1(c1_9__bhsp_3,c3_9__bhsp_3,E))) ) ) ) ) ), introduced(definition,[new_symbol(c5_9__bhsp_3),file(bhsp_3,c5_9__bhsp_3)]), [interesting(0.8),axiom,file(bhsp_3,c5_9__bhsp_3)]). fof(existence_l2_struct_0,axiom,( ? [A] : l2_struct_0(A) ), file(struct_0,l2_struct_0), [interesting(0.9),axiom,file(struct_0,l2_struct_0)]). fof(existence_l1_rlvect_1,axiom,( ? [A] : l1_rlvect_1(A) ), file(rlvect_1,l1_rlvect_1), [interesting(0.9),axiom,file(rlvect_1,l1_rlvect_1)]). 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_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(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(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_l2_rlvect_1,axiom,( ? [A] : l2_rlvect_1(A) ), file(rlvect_1,l2_rlvect_1), [interesting(0.9),axiom,file(rlvect_1,l2_rlvect_1)]). 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(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(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(existence_l1_bhsp_1,axiom,( ? [A] : l1_bhsp_1(A) ), file(bhsp_1,l1_bhsp_1), [interesting(0.9),axiom,file(bhsp_1,l1_bhsp_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(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(t1_numerals,theorem,( m1_subset_1(0,k5_numbers) ), file(numerals,t1_numerals), [interesting(0.9),axiom,file(numerals,t1_numerals)]). fof(e3_9__bhsp_3,plain,( ? [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(A,B) & r1_xreal_0(A,C) & r1_xreal_0(c4_9__bhsp_3,k5_bhsp_1(c1_9__bhsp_3,k2_normsp_1(c1_9__bhsp_3,c3_9__bhsp_3,B),k2_normsp_1(c1_9__bhsp_3,c3_9__bhsp_3,C))) ) ) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_9__bhsp_3,dt_c3_9__bhsp_3,dt_c4_9__bhsp_3,e2_9__bhsp_3,e1_9__bhsp_3])],[existence_l2_struct_0,dt_l2_struct_0,rc3_funct_1,rc4_struct_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l1_rlvect_1,dt_k1_xboole_0,dt_l1_rlvect_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_funct_1,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc6_membered,rc1_arytm_3,rc1_funct_1,rc1_membered,rc1_xreal_0,rc2_funct_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_l1_struct_0,existence_l2_rlvect_1,existence_m1_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k4_bhsp_1,dt_k5_ordinal2,dt_l1_struct_0,dt_l2_rlvect_1,dt_m1_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_funct_1,cc1_relset_1,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc5_membered,rc3_struct_0,rc5_struct_0,t1_real,t2_subset,t3_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k5_bhsp_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_l1_bhsp_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k2_normsp_1,redefinition_k5_bhsp_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_normsp_1,dt_k5_bhsp_1,dt_k5_numbers,dt_l1_bhsp_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_9__bhsp_3,dt_c3_9__bhsp_3,dt_c4_9__bhsp_3,fc2_membered,rqLessOrEqual__r1_xreal_0__r0_r0,t1_numerals,spc0_numerals,spc0_boole,e2_9__bhsp_3,e1_9__bhsp_3,d1_bhsp_3]), [interesting(0.8),file(bhsp_3,e3_9__bhsp_3),[file(bhsp_3,e3_9__bhsp_3)]]). fof(dt_c5_9__bhsp_3,plain,( m2_subset_1(c5_9__bhsp_3,k1_numbers,k5_numbers) ), inference(consider,[status(thm),assumptions([dt_c1_9__bhsp_3,dt_c3_9__bhsp_3,dt_c4_9__bhsp_3,e2_9__bhsp_3,e1_9__bhsp_3])],[dh_c5_9__bhsp_3,e3_9__bhsp_3]), [interesting(0.8),file(bhsp_3,c5_9__bhsp_3),[file(bhsp_3,c5_9__bhsp_3)]]). fof(de_c6_9__bhsp_3,definition,( c6_9__bhsp_3 = c5_9__bhsp_3 ), introduced(definition,[new_symbol(c6_9__bhsp_3),file(bhsp_3,c6_9__bhsp_3)]), [interesting(0.8),axiom,file(bhsp_3,c6_9__bhsp_3)]). fof(dt_c6_9__bhsp_3,plain,( m2_subset_1(c6_9__bhsp_3,k1_numbers,k5_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_9__bhsp_3,dt_c3_9__bhsp_3,dt_c4_9__bhsp_3,e2_9__bhsp_3,e1_9__bhsp_3])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc2_xreal_0,cc3_membered,cc7_xreal_0,fc6_membered,rc1_arytm_3,rc1_membered,rc1_xreal_0,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_funct_1,cc3_arytm_3,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc5_membered,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_c5_9__bhsp_3,fc2_membered,de_c6_9__bhsp_3]), [interesting(0.8),file(bhsp_3,c6_9__bhsp_3),[file(bhsp_3,c6_9__bhsp_3)]]). fof(dh_c7_9__bhsp_3,definition, ( ( m2_subset_1(c7_9__bhsp_3,k1_numbers,k5_numbers) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c6_9__bhsp_3,c7_9__bhsp_3) & r1_xreal_0(c6_9__bhsp_3,A) & r1_xreal_0(c4_9__bhsp_3,k5_bhsp_1(c1_9__bhsp_3,k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),c7_9__bhsp_3),k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),A))) ) ) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c6_9__bhsp_3,B) & r1_xreal_0(c6_9__bhsp_3,C) & r1_xreal_0(c4_9__bhsp_3,k5_bhsp_1(c1_9__bhsp_3,k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),B),k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),C))) ) ) ) ), introduced(definition,[new_symbol(c7_9__bhsp_3),file(bhsp_3,c7_9__bhsp_3)]), [interesting(0.8),axiom,file(bhsp_3,c7_9__bhsp_3)]). fof(dh_c8_9__bhsp_3,definition, ( ( m2_subset_1(c8_9__bhsp_3,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c6_9__bhsp_3,c7_9__bhsp_3) & r1_xreal_0(c6_9__bhsp_3,c8_9__bhsp_3) & r1_xreal_0(c4_9__bhsp_3,k5_bhsp_1(c1_9__bhsp_3,k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),c7_9__bhsp_3),k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),c8_9__bhsp_3))) ) ) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c6_9__bhsp_3,c7_9__bhsp_3) & r1_xreal_0(c6_9__bhsp_3,A) & r1_xreal_0(c4_9__bhsp_3,k5_bhsp_1(c1_9__bhsp_3,k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),c7_9__bhsp_3),k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),A))) ) ) ), introduced(definition,[new_symbol(c8_9__bhsp_3),file(bhsp_3,c8_9__bhsp_3)]), [interesting(0.8),axiom,file(bhsp_3,c8_9__bhsp_3)]). fof(e5_9__bhsp_3,assumption, ( r1_xreal_0(c6_9__bhsp_3,c7_9__bhsp_3) & r1_xreal_0(c6_9__bhsp_3,c8_9__bhsp_3) ), introduced(assumption,[file(bhsp_3,e5_9__bhsp_3)]), [interesting(0.8),axiom,file(bhsp_3,e5_9__bhsp_3)]). fof(dt_k2_rlvect_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_rlvect_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k2_rlvect_1(A,B,C),u1_struct_0(A)) ) ), file(rlvect_1,k2_rlvect_1), [interesting(0.9),axiom,file(rlvect_1,k2_rlvect_1)]). fof(commutativity_k4_rlvect_1,theorem,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & l1_rlvect_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => k4_rlvect_1(A,B,C) = k4_rlvect_1(A,C,B) ) ), file(rlvect_1,k4_rlvect_1), [interesting(0.9),axiom,file(rlvect_1,k4_rlvect_1)]). fof(redefinition_k4_rlvect_1,definition,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & l1_rlvect_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => k4_rlvect_1(A,B,C) = k2_rlvect_1(A,B,C) ) ), file(rlvect_1,k4_rlvect_1), [interesting(0.9),axiom,file(rlvect_1,k4_rlvect_1)]). fof(dt_k4_rlvect_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & l1_rlvect_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k4_rlvect_1(A,B,C),u1_struct_0(A)) ) ), file(rlvect_1,k4_rlvect_1), [interesting(0.9),axiom,file(rlvect_1,k4_rlvect_1)]). fof(dt_c7_9__bhsp_3,assumption,( m2_subset_1(c7_9__bhsp_3,k1_numbers,k5_numbers) ), introduced(assumption,[file(bhsp_3,c7_9__bhsp_3)]), [interesting(0.8),axiom,file(bhsp_3,c7_9__bhsp_3)]). fof(dt_c8_9__bhsp_3,assumption,( m2_subset_1(c8_9__bhsp_3,k1_numbers,k5_numbers) ), introduced(assumption,[file(bhsp_3,c8_9__bhsp_3)]), [interesting(0.8),axiom,file(bhsp_3,c8_9__bhsp_3)]). fof(fc10_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v2_xreal_0(k2_xcmplx_0(B,A)) & ~ v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc10_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc10_xreal_0)]). fof(fc11_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v2_xreal_0(k2_xcmplx_0(A,B)) & v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc11_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc11_xreal_0)]). fof(fc12_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & ~ v2_xreal_0(k2_xcmplx_0(B,A)) & v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc12_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc12_xreal_0)]). fof(fc7_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc7_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc7_xreal_0)]). fof(fc9_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v2_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc9_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc9_xreal_0)]). 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(dt_k2_xcmplx_0,axiom,( $true ), file(xcmplx_0,k2_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k2_xcmplx_0)]). fof(fc3_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc3_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc3_xreal_0)]). fof(fc8_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v2_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc8_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc8_xreal_0)]). 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(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(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(commutativity_k3_real_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k3_real_1(A,B) = k3_real_1(B,A) ) ), file(real_1,k3_real_1), [interesting(0.9),axiom,file(real_1,k3_real_1)]). fof(redefinition_k3_real_1,definition,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k3_real_1(A,B) = k2_xcmplx_0(A,B) ) ), file(real_1,k3_real_1), [interesting(0.9),axiom,file(real_1,k3_real_1)]). fof(dt_k3_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k3_real_1(A,B),k1_numbers) ) ), file(real_1,k3_real_1), [interesting(0.9),axiom,file(real_1,k3_real_1)]). fof(t47_bhsp_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_rlvect_1(A) & v2_bhsp_1(A) & l1_bhsp_1(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ! [E] : ( m1_subset_1(E,u1_struct_0(A)) => r1_xreal_0(k5_bhsp_1(A,k4_rlvect_1(A,B,C),k4_rlvect_1(A,D,E)),k3_real_1(k5_bhsp_1(A,B,D),k5_bhsp_1(A,C,E))) ) ) ) ) ) ), file(bhsp_1,t47_bhsp_1), [interesting(0.9),axiom,file(bhsp_1,t47_bhsp_1)]). fof(e7_9__bhsp_3,plain,( r1_xreal_0(k5_bhsp_1(c1_9__bhsp_3,k4_rlvect_1(c1_9__bhsp_3,k2_normsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c7_9__bhsp_3),c2_9__bhsp_3),k4_rlvect_1(c1_9__bhsp_3,k2_normsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c8_9__bhsp_3),c2_9__bhsp_3)),k3_real_1(k5_bhsp_1(c1_9__bhsp_3,k2_normsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c7_9__bhsp_3),k2_normsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c8_9__bhsp_3)),k5_bhsp_1(c1_9__bhsp_3,c2_9__bhsp_3,c2_9__bhsp_3))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_9__bhsp_3,dt_c2_9__bhsp_3,dt_c3_9__bhsp_3,dt_c7_9__bhsp_3,dt_c8_9__bhsp_3])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc6_membered,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_arytm_3,rc1_membered,rc2_xreal_0,rc3_funct_1,rc3_xreal_0,rc4_xreal_0,t1_real,t1_subset,t2_real,t3_real,t4_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_l2_struct_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_l2_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_funct_1,cc1_relset_1,cc2_funct_1,cc3_arytm_3,cc4_membered,cc6_membered,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc5_membered,rc1_funct_1,rc1_xreal_0,rc2_funct_1,rc4_struct_0,rc5_struct_0,spc6_arithm,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,commutativity_k2_xcmplx_0,existence_l1_rlvect_1,existence_l1_struct_0,existence_l2_rlvect_1,existence_m1_relset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k2_rlvect_1,dt_k2_xcmplx_0,dt_k4_bhsp_1,dt_k5_numbers,dt_l1_rlvect_1,dt_l1_struct_0,dt_l2_rlvect_1,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,cc2_xreal_0,fc1_struct_0,fc2_membered,fc3_xreal_0,rc3_struct_0,commutativity_k3_real_1,commutativity_k4_rlvect_1,commutativity_k5_bhsp_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_l1_bhsp_1,existence_m1_subset_1,redefinition_k2_normsp_1,redefinition_k3_real_1,redefinition_k4_rlvect_1,redefinition_k5_bhsp_1,dt_k2_normsp_1,dt_k3_real_1,dt_k4_rlvect_1,dt_k5_bhsp_1,dt_l1_bhsp_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_9__bhsp_3,dt_c2_9__bhsp_3,dt_c3_9__bhsp_3,dt_c7_9__bhsp_3,dt_c8_9__bhsp_3,t47_bhsp_1]), [interesting(0.8),file(bhsp_3,e7_9__bhsp_3),[file(bhsp_3,e7_9__bhsp_3)]]). fof(t41_bhsp_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_rlvect_1(A) & v2_bhsp_1(A) & l1_bhsp_1(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => k5_bhsp_1(A,B,B) = 0 ) ) ), file(bhsp_1,t41_bhsp_1), [interesting(0.9),axiom,file(bhsp_1,t41_bhsp_1)]). fof(e8_9__bhsp_3,plain,( r1_xreal_0(k5_bhsp_1(c1_9__bhsp_3,k4_rlvect_1(c1_9__bhsp_3,k2_normsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c7_9__bhsp_3),c2_9__bhsp_3),k4_rlvect_1(c1_9__bhsp_3,k2_normsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c8_9__bhsp_3),c2_9__bhsp_3)),k3_real_1(k5_bhsp_1(c1_9__bhsp_3,k2_normsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c7_9__bhsp_3),k2_normsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c8_9__bhsp_3)),0)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_9__bhsp_3,dt_c2_9__bhsp_3,dt_c3_9__bhsp_3,dt_c7_9__bhsp_3,dt_c8_9__bhsp_3])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,rc1_arytm_3,rc3_funct_1,antisymmetry_r2_hidden,existence_l2_struct_0,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_l2_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_ordinal2,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_funct_1,rc1_membered,rc1_xreal_0,rc2_funct_1,rc2_xreal_0,rc3_xreal_0,rc4_struct_0,rc4_xreal_0,rc5_struct_0,spc6_arithm,t1_arithm,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,existence_l1_rlvect_1,existence_l1_struct_0,existence_l2_rlvect_1,existence_m1_relset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k2_rlvect_1,dt_k2_xcmplx_0,dt_k4_bhsp_1,dt_k5_numbers,dt_l1_rlvect_1,dt_l1_struct_0,dt_l2_rlvect_1,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,cc15_membered,cc1_funct_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_struct_0,fc2_membered,fc3_xreal_0,fc8_xreal_0,rc3_struct_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k3_real_1,commutativity_k4_rlvect_1,commutativity_k5_bhsp_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_l1_bhsp_1,existence_m1_subset_1,redefinition_k2_normsp_1,redefinition_k3_real_1,redefinition_k4_rlvect_1,redefinition_k5_bhsp_1,dt_k2_normsp_1,dt_k3_real_1,dt_k4_rlvect_1,dt_k5_bhsp_1,dt_l1_bhsp_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_9__bhsp_3,dt_c2_9__bhsp_3,dt_c3_9__bhsp_3,dt_c7_9__bhsp_3,dt_c8_9__bhsp_3,rqLessOrEqual__r1_xreal_0__r0_r0,spc0_numerals,spc0_boole,e7_9__bhsp_3,t41_bhsp_1]), [interesting(0.8),file(bhsp_3,e8_9__bhsp_3),[file(bhsp_3,e8_9__bhsp_3)]]). fof(e4_9__bhsp_3,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c5_9__bhsp_3,A) & r1_xreal_0(c5_9__bhsp_3,B) & r1_xreal_0(c4_9__bhsp_3,k5_bhsp_1(c1_9__bhsp_3,k2_normsp_1(c1_9__bhsp_3,c3_9__bhsp_3,A),k2_normsp_1(c1_9__bhsp_3,c3_9__bhsp_3,B))) ) ) ) ), inference(consider,[status(thm),assumptions([dt_c1_9__bhsp_3,dt_c3_9__bhsp_3,dt_c4_9__bhsp_3,e2_9__bhsp_3,e1_9__bhsp_3])],[dh_c5_9__bhsp_3,e3_9__bhsp_3]), [interesting(0.8),file(bhsp_3,e4_9__bhsp_3),[file(bhsp_3,e4_9__bhsp_3)]]). fof(e6_9__bhsp_3,plain,( ~ r1_xreal_0(c4_9__bhsp_3,k5_bhsp_1(c1_9__bhsp_3,k2_normsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c7_9__bhsp_3),k2_normsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c8_9__bhsp_3))) ), inference(mizar_by,[status(thm),assumptions([dt_c7_9__bhsp_3,dt_c8_9__bhsp_3,e5_9__bhsp_3,dt_c1_9__bhsp_3,dt_c3_9__bhsp_3,dt_c4_9__bhsp_3,e2_9__bhsp_3,e1_9__bhsp_3])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,rc3_funct_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l2_rlvect_1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_l2_rlvect_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_funct_1,cc2_membered,cc3_membered,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc6_membered,rc1_arytm_3,rc1_funct_1,rc1_membered,rc1_xreal_0,rc2_funct_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_real,t1_subset,t2_real,t3_real,t4_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k4_bhsp_1,dt_k5_ordinal2,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_funct_1,cc2_xreal_0,cc3_arytm_3,cc4_membered,cc6_membered,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc5_membered,rc3_struct_0,rc5_struct_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,commutativity_k5_bhsp_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m2_subset_1,redefinition_k2_normsp_1,redefinition_k5_bhsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_normsp_1,dt_k5_bhsp_1,dt_k5_numbers,dt_m2_subset_1,dt_c1_9__bhsp_3,dt_c3_9__bhsp_3,dt_c4_9__bhsp_3,dt_c5_9__bhsp_3,dt_c6_9__bhsp_3,dt_c7_9__bhsp_3,dt_c8_9__bhsp_3,de_c6_9__bhsp_3,fc2_membered,e5_9__bhsp_3,e4_9__bhsp_3]), [interesting(0.8),file(bhsp_3,e6_9__bhsp_3),[file(bhsp_3,e6_9__bhsp_3)]]). fof(t2_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ( r1_xreal_0(A,B) & r1_xreal_0(B,C) ) => r1_xreal_0(A,C) ) ) ) ) ), file(xreal_1,t2_xreal_1), [interesting(0.9),axiom,file(xreal_1,t2_xreal_1)]). fof(e9_9__bhsp_3,plain,( ~ r1_xreal_0(c4_9__bhsp_3,k5_bhsp_1(c1_9__bhsp_3,k4_rlvect_1(c1_9__bhsp_3,k2_normsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c7_9__bhsp_3),c2_9__bhsp_3),k4_rlvect_1(c1_9__bhsp_3,k2_normsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c8_9__bhsp_3),c2_9__bhsp_3))) ), inference(mizar_by,[status(thm),assumptions([dt_c2_9__bhsp_3,dt_c7_9__bhsp_3,dt_c8_9__bhsp_3,e5_9__bhsp_3,dt_c1_9__bhsp_3,dt_c3_9__bhsp_3,dt_c4_9__bhsp_3,e2_9__bhsp_3,e1_9__bhsp_3])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,rc1_arytm_3,rc3_funct_1,antisymmetry_r2_hidden,existence_l2_rlvect_1,existence_l2_struct_0,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_l2_rlvect_1,dt_l2_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_ordinal2,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_funct_1,rc1_membered,rc2_funct_1,rc2_xreal_0,rc3_xreal_0,rc4_struct_0,rc4_xreal_0,rc5_struct_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,existence_l1_bhsp_1,existence_l1_rlvect_1,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k2_rlvect_1,dt_k2_xcmplx_0,dt_k4_bhsp_1,dt_k5_numbers,dt_l1_bhsp_1,dt_l1_rlvect_1,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_funct_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_struct_0,fc2_membered,fc3_xreal_0,fc8_xreal_0,rc1_xreal_0,rc3_struct_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,spc6_arithm,t1_arithm,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k3_real_1,commutativity_k4_rlvect_1,commutativity_k5_bhsp_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k2_normsp_1,redefinition_k3_real_1,redefinition_k4_rlvect_1,redefinition_k5_bhsp_1,dt_k2_normsp_1,dt_k3_real_1,dt_k4_rlvect_1,dt_k5_bhsp_1,dt_c1_9__bhsp_3,dt_c2_9__bhsp_3,dt_c3_9__bhsp_3,dt_c4_9__bhsp_3,dt_c7_9__bhsp_3,dt_c8_9__bhsp_3,cc2_xreal_0,spc0_numerals,spc0_boole,e8_9__bhsp_3,e6_9__bhsp_3,t2_xreal_1,rqLessOrEqual__r1_xreal_0__r0_r0]), [interesting(0.8),file(bhsp_3,e9_9__bhsp_3),[file(bhsp_3,e9_9__bhsp_3)]]). fof(d12_bhsp_1,definition,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_rlvect_1(A) & v2_bhsp_1(A) & l1_bhsp_1(A) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(A)) & m2_relset_1(B,k5_numbers,u1_struct_0(A)) ) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k5_numbers,u1_struct_0(A)) & m2_relset_1(D,k5_numbers,u1_struct_0(A)) ) => ( D = k7_bhsp_1(A,B,C) <=> ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => k2_normsp_1(A,D,E) = k4_rlvect_1(A,k2_normsp_1(A,B,E),C) ) ) ) ) ) ) ), file(bhsp_1,d12_bhsp_1), [interesting(0.9),axiom,file(bhsp_1,d12_bhsp_1)]). fof(e10_9__bhsp_3,plain,( ~ r1_xreal_0(c4_9__bhsp_3,k5_bhsp_1(c1_9__bhsp_3,k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),c7_9__bhsp_3),k4_rlvect_1(c1_9__bhsp_3,k2_normsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c8_9__bhsp_3),c2_9__bhsp_3))) ), inference(mizar_by,[status(thm),assumptions([dt_c2_9__bhsp_3,dt_c7_9__bhsp_3,dt_c8_9__bhsp_3,e5_9__bhsp_3,dt_c1_9__bhsp_3,dt_c3_9__bhsp_3,dt_c4_9__bhsp_3,e2_9__bhsp_3,e1_9__bhsp_3])],[rc3_funct_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l2_struct_0,dt_k1_xboole_0,dt_l2_struct_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_funct_1,cc2_membered,cc3_membered,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc6_membered,rc1_arytm_3,rc1_funct_1,rc1_membered,rc1_xreal_0,rc2_funct_1,rc2_xreal_0,rc3_xreal_0,rc4_struct_0,rc4_xreal_0,t1_real,t1_subset,t2_real,t3_real,t4_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_l1_rlvect_1,existence_l1_struct_0,existence_l2_rlvect_1,existence_m1_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_rlvect_1,dt_k2_zfmisc_1,dt_k4_bhsp_1,dt_k5_ordinal2,dt_l1_rlvect_1,dt_l1_struct_0,dt_l2_rlvect_1,dt_m1_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_funct_1,cc1_relset_1,cc2_xreal_0,cc3_arytm_3,cc4_membered,cc6_membered,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc5_membered,rc3_struct_0,rc5_struct_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,commutativity_k4_rlvect_1,commutativity_k5_bhsp_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_l1_bhsp_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k2_normsp_1,redefinition_k4_rlvect_1,redefinition_k5_bhsp_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_normsp_1,dt_k4_rlvect_1,dt_k5_bhsp_1,dt_k5_numbers,dt_k7_bhsp_1,dt_l1_bhsp_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_9__bhsp_3,dt_c2_9__bhsp_3,dt_c3_9__bhsp_3,dt_c4_9__bhsp_3,dt_c7_9__bhsp_3,dt_c8_9__bhsp_3,fc2_membered,e9_9__bhsp_3,d12_bhsp_1]), [interesting(0.8),file(bhsp_3,e10_9__bhsp_3),[file(bhsp_3,e10_9__bhsp_3)]]). fof(e11_9__bhsp_3,plain,( ~ r1_xreal_0(c4_9__bhsp_3,k5_bhsp_1(c1_9__bhsp_3,k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),c7_9__bhsp_3),k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),c8_9__bhsp_3))) ), inference(mizar_by,[status(thm),assumptions([dt_c2_9__bhsp_3,dt_c7_9__bhsp_3,dt_c8_9__bhsp_3,e5_9__bhsp_3,dt_c1_9__bhsp_3,dt_c3_9__bhsp_3,dt_c4_9__bhsp_3,e2_9__bhsp_3,e1_9__bhsp_3])],[rc3_funct_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l2_struct_0,dt_k1_xboole_0,dt_l2_struct_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_funct_1,cc2_membered,cc3_membered,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc6_membered,rc1_arytm_3,rc1_funct_1,rc1_membered,rc1_xreal_0,rc2_funct_1,rc2_xreal_0,rc3_xreal_0,rc4_struct_0,rc4_xreal_0,t1_real,t1_subset,t2_real,t3_real,t4_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_l1_rlvect_1,existence_l1_struct_0,existence_l2_rlvect_1,existence_m1_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_rlvect_1,dt_k2_zfmisc_1,dt_k4_bhsp_1,dt_k5_ordinal2,dt_l1_rlvect_1,dt_l1_struct_0,dt_l2_rlvect_1,dt_m1_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_funct_1,cc1_relset_1,cc2_xreal_0,cc3_arytm_3,cc4_membered,cc6_membered,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc5_membered,rc3_struct_0,rc5_struct_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,commutativity_k4_rlvect_1,commutativity_k5_bhsp_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_l1_bhsp_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k2_normsp_1,redefinition_k4_rlvect_1,redefinition_k5_bhsp_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_normsp_1,dt_k4_rlvect_1,dt_k5_bhsp_1,dt_k5_numbers,dt_k7_bhsp_1,dt_l1_bhsp_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_9__bhsp_3,dt_c2_9__bhsp_3,dt_c3_9__bhsp_3,dt_c4_9__bhsp_3,dt_c7_9__bhsp_3,dt_c8_9__bhsp_3,fc2_membered,e10_9__bhsp_3,d12_bhsp_1]), [interesting(0.8),file(bhsp_3,e11_9__bhsp_3),[file(bhsp_3,e11_9__bhsp_3)]]). fof(i11_9__bhsp_3,theorem,( $true ), introduced(tautology,[file(bhsp_3,i11_9__bhsp_3)]), [interesting(0.8),trivial,file(bhsp_3,i11_9__bhsp_3)]). fof(i10_9__bhsp_3,plain,( ~ r1_xreal_0(c4_9__bhsp_3,k5_bhsp_1(c1_9__bhsp_3,k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),c7_9__bhsp_3),k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),c8_9__bhsp_3))) ), inference(conclusion,[status(thm),assumptions([dt_c2_9__bhsp_3,dt_c7_9__bhsp_3,dt_c8_9__bhsp_3,e5_9__bhsp_3,dt_c1_9__bhsp_3,dt_c3_9__bhsp_3,dt_c4_9__bhsp_3,e2_9__bhsp_3,e1_9__bhsp_3])],[e11_9__bhsp_3,i11_9__bhsp_3]), [interesting(0.8),file(bhsp_3,i10_9__bhsp_3),[file(bhsp_3,i10_9__bhsp_3)]]). fof(i9_9__bhsp_3,plain,( ~ ( r1_xreal_0(c6_9__bhsp_3,c7_9__bhsp_3) & r1_xreal_0(c6_9__bhsp_3,c8_9__bhsp_3) & r1_xreal_0(c4_9__bhsp_3,k5_bhsp_1(c1_9__bhsp_3,k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),c7_9__bhsp_3),k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),c8_9__bhsp_3))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_9__bhsp_3,dt_c7_9__bhsp_3,dt_c8_9__bhsp_3,dt_c1_9__bhsp_3,dt_c3_9__bhsp_3,dt_c4_9__bhsp_3,e2_9__bhsp_3,e1_9__bhsp_3]),discharge_asm(discharge,[e5_9__bhsp_3])],[e5_9__bhsp_3,i10_9__bhsp_3]), [interesting(0.8),file(bhsp_3,i9_9__bhsp_3),[file(bhsp_3,i9_9__bhsp_3)]]). fof(i9_9_tmp__bhsp_3,plain, ( m2_subset_1(c8_9__bhsp_3,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c6_9__bhsp_3,c7_9__bhsp_3) & r1_xreal_0(c6_9__bhsp_3,c8_9__bhsp_3) & r1_xreal_0(c4_9__bhsp_3,k5_bhsp_1(c1_9__bhsp_3,k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),c7_9__bhsp_3),k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),c8_9__bhsp_3))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_9__bhsp_3,dt_c7_9__bhsp_3,dt_c1_9__bhsp_3,dt_c3_9__bhsp_3,dt_c4_9__bhsp_3,e2_9__bhsp_3,e1_9__bhsp_3]),discharge_asm(discharge,[dt_c8_9__bhsp_3])],[dt_c8_9__bhsp_3,i9_9__bhsp_3]), [interesting(0.8),i8_9__bhsp_3]). fof(i8_9__bhsp_3,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c6_9__bhsp_3,c7_9__bhsp_3) & r1_xreal_0(c6_9__bhsp_3,A) & r1_xreal_0(c4_9__bhsp_3,k5_bhsp_1(c1_9__bhsp_3,k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),c7_9__bhsp_3),k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),A))) ) ) ), inference(let,[status(thm),assumptions([dt_c2_9__bhsp_3,dt_c7_9__bhsp_3,dt_c1_9__bhsp_3,dt_c3_9__bhsp_3,dt_c4_9__bhsp_3,e2_9__bhsp_3,e1_9__bhsp_3])],[i9_9_tmp__bhsp_3,dh_c8_9__bhsp_3]), [interesting(0.8),file(bhsp_3,i8_9__bhsp_3),[file(bhsp_3,i8_9__bhsp_3)]]). fof(i8_9_tmp__bhsp_3,plain, ( m2_subset_1(c7_9__bhsp_3,k1_numbers,k5_numbers) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c6_9__bhsp_3,c7_9__bhsp_3) & r1_xreal_0(c6_9__bhsp_3,A) & r1_xreal_0(c4_9__bhsp_3,k5_bhsp_1(c1_9__bhsp_3,k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),c7_9__bhsp_3),k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),A))) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_9__bhsp_3,dt_c1_9__bhsp_3,dt_c3_9__bhsp_3,dt_c4_9__bhsp_3,e2_9__bhsp_3,e1_9__bhsp_3]),discharge_asm(discharge,[dt_c7_9__bhsp_3])],[dt_c7_9__bhsp_3,i8_9__bhsp_3]), [interesting(0.8),i7_9__bhsp_3]). fof(i7_9__bhsp_3,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c6_9__bhsp_3,A) & r1_xreal_0(c6_9__bhsp_3,B) & r1_xreal_0(c4_9__bhsp_3,k5_bhsp_1(c1_9__bhsp_3,k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),A),k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),B))) ) ) ) ), inference(let,[status(thm),assumptions([dt_c2_9__bhsp_3,dt_c1_9__bhsp_3,dt_c3_9__bhsp_3,dt_c4_9__bhsp_3,e2_9__bhsp_3,e1_9__bhsp_3])],[i8_9_tmp__bhsp_3,dh_c7_9__bhsp_3]), [interesting(0.8),file(bhsp_3,i7_9__bhsp_3),[file(bhsp_3,i7_9__bhsp_3)]]). fof(i6_9__bhsp_3,plain,( ? [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(A,B) & r1_xreal_0(A,C) & r1_xreal_0(c4_9__bhsp_3,k5_bhsp_1(c1_9__bhsp_3,k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),B),k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),C))) ) ) ) ) ), inference(take,[status(thm),assumptions([dt_c2_9__bhsp_3,dt_c1_9__bhsp_3,dt_c3_9__bhsp_3,dt_c4_9__bhsp_3,e2_9__bhsp_3,e1_9__bhsp_3])],[dt_l2_struct_0,rc4_struct_0,dt_l1_rlvect_1,rc3_funct_1,dt_k2_zfmisc_1,dt_l2_rlvect_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_funct_1,cc2_membered,cc3_membered,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_arytm_3,rc1_funct_1,rc1_membered,rc1_xreal_0,rc2_funct_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k4_bhsp_1,dt_k5_ordinal2,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_funct_1,cc2_xreal_0,cc3_arytm_3,cc4_membered,cc6_membered,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc5_membered,rc3_struct_0,rc5_struct_0,commutativity_k5_bhsp_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k2_normsp_1,redefinition_k5_bhsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_normsp_1,dt_k5_bhsp_1,dt_k5_numbers,dt_k7_bhsp_1,dt_m2_subset_1,dt_c1_9__bhsp_3,dt_c2_9__bhsp_3,dt_c3_9__bhsp_3,dt_c4_9__bhsp_3,dt_c6_9__bhsp_3,fc2_membered,i7_9__bhsp_3]), [interesting(0.8),file(bhsp_3,i6_9__bhsp_3),[file(bhsp_3,i6_9__bhsp_3)]]). fof(i5_9__bhsp_3,plain,( ~ ( ~ r1_xreal_0(c4_9__bhsp_3,0) & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ? [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) & ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & r1_xreal_0(A,B) & r1_xreal_0(A,C) & r1_xreal_0(c4_9__bhsp_3,k5_bhsp_1(c1_9__bhsp_3,k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),B),k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),C))) ) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_9__bhsp_3,dt_c1_9__bhsp_3,dt_c3_9__bhsp_3,dt_c4_9__bhsp_3,e1_9__bhsp_3]),discharge_asm(discharge,[e2_9__bhsp_3])],[e2_9__bhsp_3,i6_9__bhsp_3]), [interesting(0.8),file(bhsp_3,i5_9__bhsp_3),[file(bhsp_3,i5_9__bhsp_3)]]). fof(i5_9_tmp__bhsp_3,plain, ( m1_subset_1(c4_9__bhsp_3,k1_numbers) => ~ ( ~ r1_xreal_0(c4_9__bhsp_3,0) & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ? [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) & ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & r1_xreal_0(A,B) & r1_xreal_0(A,C) & r1_xreal_0(c4_9__bhsp_3,k5_bhsp_1(c1_9__bhsp_3,k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),B),k2_normsp_1(c1_9__bhsp_3,k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),C))) ) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_9__bhsp_3,dt_c1_9__bhsp_3,dt_c3_9__bhsp_3,e1_9__bhsp_3]),discharge_asm(discharge,[dt_c4_9__bhsp_3])],[dt_c4_9__bhsp_3,i5_9__bhsp_3]), [interesting(0.8),i4_9__bhsp_3]). fof(i4_9__bhsp_3,plain,( v1_bhsp_3(k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),c1_9__bhsp_3) ), inference(let,[status(thm),assumptions([dt_c2_9__bhsp_3,dt_c1_9__bhsp_3,dt_c3_9__bhsp_3,e1_9__bhsp_3])],[i5_9_tmp__bhsp_3,dt_l2_struct_0,rc3_funct_1,rc4_struct_0,dt_l1_rlvect_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_funct_1,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,rc1_arytm_3,rc1_funct_1,rc1_membered,rc1_xreal_0,rc2_funct_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k4_bhsp_1,dt_k5_ordinal2,dt_l1_struct_0,dt_l2_rlvect_1,dt_m1_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_funct_1,cc1_relset_1,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc5_membered,rc3_struct_0,rc5_struct_0,commutativity_k5_bhsp_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k2_normsp_1,redefinition_k5_bhsp_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_normsp_1,dt_k5_bhsp_1,dt_k5_numbers,dt_k7_bhsp_1,dt_l1_bhsp_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_9__bhsp_3,dt_c2_9__bhsp_3,dt_c3_9__bhsp_3,fc2_membered,spc0_numerals,spc0_boole,d1_bhsp_3,dh_c4_9__bhsp_3]), [interesting(0.8),file(bhsp_3,i4_9__bhsp_3),[file(bhsp_3,i4_9__bhsp_3)]]). fof(i3_9__bhsp_3,plain, ( v1_bhsp_3(c3_9__bhsp_3,c1_9__bhsp_3) => v1_bhsp_3(k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),c1_9__bhsp_3) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_9__bhsp_3,dt_c1_9__bhsp_3,dt_c3_9__bhsp_3]),discharge_asm(discharge,[e1_9__bhsp_3])],[e1_9__bhsp_3,i4_9__bhsp_3]), [interesting(0.8),file(bhsp_3,i3_9__bhsp_3),[file(bhsp_3,i3_9__bhsp_3)]]). fof(i3_9_tmp__bhsp_3,plain, ( ( v1_funct_1(c3_9__bhsp_3) & v1_funct_2(c3_9__bhsp_3,k5_numbers,u1_struct_0(c1_9__bhsp_3)) & m2_relset_1(c3_9__bhsp_3,k5_numbers,u1_struct_0(c1_9__bhsp_3)) ) => ( v1_bhsp_3(c3_9__bhsp_3,c1_9__bhsp_3) => v1_bhsp_3(k7_bhsp_1(c1_9__bhsp_3,c3_9__bhsp_3,c2_9__bhsp_3),c1_9__bhsp_3) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_9__bhsp_3,dt_c1_9__bhsp_3]),discharge_asm(discharge,[dt_c3_9__bhsp_3])],[dt_c3_9__bhsp_3,i3_9__bhsp_3]), [interesting(0.8),i2_9__bhsp_3]). fof(i2_9__bhsp_3,plain,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,u1_struct_0(c1_9__bhsp_3)) & m2_relset_1(A,k5_numbers,u1_struct_0(c1_9__bhsp_3)) ) => ( v1_bhsp_3(A,c1_9__bhsp_3) => v1_bhsp_3(k7_bhsp_1(c1_9__bhsp_3,A,c2_9__bhsp_3),c1_9__bhsp_3) ) ) ), inference(let,[status(thm),assumptions([dt_c2_9__bhsp_3,dt_c1_9__bhsp_3])],[i3_9_tmp__bhsp_3,dh_c3_9__bhsp_3]), [interesting(0.8),file(bhsp_3,i2_9__bhsp_3),[file(bhsp_3,i2_9__bhsp_3)]]). fof(i2_9_tmp__bhsp_3,plain, ( m1_subset_1(c2_9__bhsp_3,u1_struct_0(c1_9__bhsp_3)) => ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,u1_struct_0(c1_9__bhsp_3)) & m2_relset_1(A,k5_numbers,u1_struct_0(c1_9__bhsp_3)) ) => ( v1_bhsp_3(A,c1_9__bhsp_3) => v1_bhsp_3(k7_bhsp_1(c1_9__bhsp_3,A,c2_9__bhsp_3),c1_9__bhsp_3) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_9__bhsp_3]),discharge_asm(discharge,[dt_c2_9__bhsp_3])],[dt_c2_9__bhsp_3,i2_9__bhsp_3]), [interesting(0.8),i1_9__bhsp_3]). fof(i1_9__bhsp_3,plain,( ! [A] : ( m1_subset_1(A,u1_struct_0(c1_9__bhsp_3)) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(c1_9__bhsp_3)) & m2_relset_1(B,k5_numbers,u1_struct_0(c1_9__bhsp_3)) ) => ( v1_bhsp_3(B,c1_9__bhsp_3) => v1_bhsp_3(k7_bhsp_1(c1_9__bhsp_3,B,A),c1_9__bhsp_3) ) ) ) ), inference(let,[status(thm),assumptions([dt_c1_9__bhsp_3])],[i2_9_tmp__bhsp_3,dh_c2_9__bhsp_3]), [interesting(0.8),file(bhsp_3,i1_9__bhsp_3),[file(bhsp_3,i1_9__bhsp_3)]]). fof(i1_9_tmp__bhsp_3,plain, ( ( ~ v3_struct_0(c1_9__bhsp_3) & v3_rlvect_1(c1_9__bhsp_3) & v4_rlvect_1(c1_9__bhsp_3) & v5_rlvect_1(c1_9__bhsp_3) & v6_rlvect_1(c1_9__bhsp_3) & v7_rlvect_1(c1_9__bhsp_3) & v2_bhsp_1(c1_9__bhsp_3) & l1_bhsp_1(c1_9__bhsp_3) ) => ! [A] : ( m1_subset_1(A,u1_struct_0(c1_9__bhsp_3)) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(c1_9__bhsp_3)) & m2_relset_1(B,k5_numbers,u1_struct_0(c1_9__bhsp_3)) ) => ( v1_bhsp_3(B,c1_9__bhsp_3) => v1_bhsp_3(k7_bhsp_1(c1_9__bhsp_3,B,A),c1_9__bhsp_3) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_9__bhsp_3])],[dt_c1_9__bhsp_3,i1_9__bhsp_3]), [interesting(1),t7_bhsp_3]). fof(t7_bhsp_3,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_rlvect_1(A) & v2_bhsp_1(A) & l1_bhsp_1(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,u1_struct_0(A)) & m2_relset_1(C,k5_numbers,u1_struct_0(A)) ) => ( v1_bhsp_3(C,A) => v1_bhsp_3(k7_bhsp_1(A,C,B),A) ) ) ) ) ), inference(let,[status(thm),assumptions([])],[i1_9_tmp__bhsp_3,dh_c1_9__bhsp_3]), [interesting(1),file(bhsp_3,t7_bhsp_3),[file(bhsp_3,t7_bhsp_3)]]).