SET007 Axioms: SET007+838.ax
%------------------------------------------------------------------------------
% File : SET007+838 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Fundamental Group of Convex Subspaces of cal E^n_T
% Version : [Urb08] axioms.
% English :
% Refs : [Mat90] Matuszewski (1990), Formalized Mathematics
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : topalg_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 48 ( 2 unt; 0 def)
% Number of atoms : 324 ( 20 equ)
% Maximal formula atoms : 19 ( 6 avg)
% Number of connectives : 306 ( 30 ~; 0 |; 146 &)
% ( 6 <=>; 124 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 9 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 35 ( 34 usr; 0 prp; 1-6 aty)
% Number of functors : 37 ( 37 usr; 7 con; 0-6 aty)
% Number of variables : 157 ( 152 !; 5 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(rc1_topalg_2,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
& ~ v1_xboole_0(B)
& v1_jordan1(B,A) ) ) ).
fof(cc1_topalg_2,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ! [B] :
( m1_pre_topc(B,k15_euclid(A))
=> ( ( ~ v3_struct_0(B)
& v1_topalg_2(B,A) )
=> ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& v1_borsuk_2(B)
& v1_connsp_1(B) ) ) ) ) ).
fof(rc2_topalg_2,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ? [B] :
( m1_pre_topc(B,k15_euclid(A))
& ~ v3_struct_0(B)
& v1_pre_topc(B)
& v2_pre_topc(B)
& v1_borsuk_2(B)
& v1_connsp_1(B)
& v1_topalg_2(B,A) ) ) ).
fof(cc2_topalg_2,axiom,
! [A,B,C,D,E,F] :
( ( m1_subset_1(A,k5_numbers)
& ~ v3_struct_0(B)
& v1_topalg_2(B,A)
& m1_pre_topc(B,k15_euclid(A))
& m1_subset_1(C,u1_struct_0(B))
& m1_subset_1(D,u1_struct_0(B))
& m1_borsuk_2(E,B,C,D)
& m1_borsuk_2(F,B,C,D) )
=> ! [G] :
( m1_borsuk_6(G,B,C,D,E,F)
=> ( v1_relat_1(G)
& v5_pre_topc(G,k6_borsuk_1(k5_topmetr,k5_topmetr),B) ) ) ) ).
fof(fc1_topalg_2,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& ~ v3_struct_0(B)
& v1_topalg_2(B,A)
& m1_pre_topc(B,k15_euclid(A))
& m1_subset_1(C,u1_struct_0(B)) )
=> ( ~ v3_struct_0(k3_topalg_1(B,C))
& v1_group_1(k3_topalg_1(B,C))
& v2_group_1(k3_topalg_1(B,C))
& v3_group_1(k3_topalg_1(B,C))
& v4_group_1(k3_topalg_1(B,C))
& v3_realset2(k3_topalg_1(B,C)) ) ) ).
fof(cc3_topalg_2,axiom,
! [A] :
( m1_pre_topc(A,k3_topmetr)
=> ( v2_pre_topc(A)
& v1_borsuk_6(A) ) ) ).
fof(rc3_topalg_2,axiom,
? [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
& ~ v1_xboole_0(A)
& v2_topalg_2(A) ) ).
fof(cc4_topalg_2,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
=> ( v1_xboole_0(A)
=> v2_topalg_2(A) ) ) ).
fof(rc4_topalg_2,axiom,
? [A] :
( m1_pre_topc(A,k3_topmetr)
& ~ v3_struct_0(A)
& v1_pre_topc(A)
& v2_pre_topc(A)
& v1_borsuk_6(A)
& v3_topalg_2(A) ) ).
fof(cc5_topalg_2,axiom,
! [A] :
( m1_pre_topc(A,k3_topalg_2)
=> ( ( ~ v3_struct_0(A)
& v3_topalg_2(A) )
=> ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& v1_borsuk_2(A)
& v1_borsuk_6(A)
& v1_connsp_1(A) ) ) ) ).
fof(cc6_topalg_2,axiom,
! [A,B,C,D,E] :
( ( ~ v3_struct_0(A)
& v3_topalg_2(A)
& m1_pre_topc(A,k3_topalg_2)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A))
& m1_borsuk_2(D,A,B,C)
& m1_borsuk_2(E,A,B,C) )
=> ! [F] :
( m1_borsuk_6(F,A,B,C,D,E)
=> ( v1_relat_1(F)
& v5_pre_topc(F,k6_borsuk_1(k5_topmetr,k5_topmetr),A) ) ) ) ).
fof(fc2_topalg_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_topalg_2(A)
& m1_pre_topc(A,k3_topalg_2)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( ~ v3_struct_0(k3_topalg_1(A,B))
& v1_group_1(k3_topalg_1(A,B))
& v2_group_1(k3_topalg_1(A,B))
& v3_group_1(k3_topalg_1(A,B))
& v4_group_1(k3_topalg_1(A,B))
& v3_realset2(k3_topalg_1(A,B)) ) ) ).
fof(fc3_topalg_2,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k5_topmetr))
=> ( ~ v3_struct_0(k3_topalg_1(k5_topmetr,A))
& v1_group_1(k3_topalg_1(k5_topmetr,A))
& v2_group_1(k3_topalg_1(k5_topmetr,A))
& v3_group_1(k3_topalg_1(k5_topmetr,A))
& v4_group_1(k3_topalg_1(k5_topmetr,A))
& v3_realset2(k3_topalg_1(k5_topmetr,A)) ) ) ).
fof(d1_topalg_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_pre_topc(B,k15_euclid(A))
=> ( v1_topalg_2(B,A)
<=> ( v1_jordan1(k2_pre_topc(B),A)
& m1_subset_1(k2_pre_topc(B),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ) ) ) ).
fof(t1_topalg_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_pre_topc(B,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(B))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ! [G] :
( m1_borsuk_2(G,B,E,F)
=> ( ( C = E
& D = F
& r1_borsuk_6(B,E,F) )
=> m1_borsuk_2(G,A,C,D) ) ) ) ) ) ) ) ) ).
fof(d2_topalg_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_topalg_2(B,A)
& m1_pre_topc(B,k15_euclid(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m1_borsuk_2(E,B,C,D)
=> ! [F] :
( m1_borsuk_2(F,B,C,D)
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,u1_struct_0(k6_borsuk_1(k5_topmetr,k5_topmetr)),u1_struct_0(B))
& m2_relset_1(G,u1_struct_0(k6_borsuk_1(k5_topmetr,k5_topmetr)),u1_struct_0(B)) )
=> ( G = k1_topalg_2(A,B,C,D,E,F)
<=> ! [H] :
( m1_subset_1(H,u1_struct_0(k5_topmetr))
=> ! [I] :
( m1_subset_1(I,u1_struct_0(k5_topmetr))
=> ! [J] :
( m1_subset_1(J,u1_struct_0(k15_euclid(A)))
=> ! [K] :
( m1_subset_1(K,u1_struct_0(k15_euclid(A)))
=> ( ( J = k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(B),E,H)
& K = k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(B),F,H) )
=> k1_binop_1(G,H,I) = k17_euclid(A,k18_euclid(k6_xcmplx_0(np__1,I),A,J),k18_euclid(I,A,K)) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t2_topalg_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_topalg_2(B,A)
& m1_pre_topc(B,k15_euclid(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m1_borsuk_2(E,B,C,D)
=> ! [F] :
( m1_borsuk_2(F,B,C,D)
=> r4_borsuk_2(B,C,D,E,F) ) ) ) ) ) ) ).
fof(t3_topalg_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_topalg_2(B,A)
& m1_pre_topc(B,k15_euclid(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_borsuk_2(D,B,C,C)
=> u1_struct_0(k3_topalg_1(B,C)) = k1_tarski(k6_eqrel_1(k1_topalg_1(B,C),k2_topalg_1(B,C),D)) ) ) ) ) ).
fof(t4_topalg_2,axiom,
! [A] :
( v1_xreal_0(A)
=> k1_jordan2b(np__1,np__1,k2_jordan2b(A)) = A ) ).
fof(t6_topalg_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,u1_struct_0(k6_borsuk_1(k3_topmetr,k5_topmetr)),u1_struct_0(k3_topmetr))
& m2_relset_1(A,u1_struct_0(k6_borsuk_1(k3_topmetr,k5_topmetr)),u1_struct_0(k3_topmetr)) )
=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(k3_topmetr))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k5_topmetr))
=> k1_binop_1(A,B,C) = k3_xcmplx_0(k6_xcmplx_0(np__1,C),B) ) )
=> v5_pre_topc(A,k6_borsuk_1(k3_topmetr,k5_topmetr),k3_topmetr) ) ) ).
fof(t7_topalg_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,u1_struct_0(k6_borsuk_1(k3_topmetr,k5_topmetr)),u1_struct_0(k3_topmetr))
& m2_relset_1(A,u1_struct_0(k6_borsuk_1(k3_topmetr,k5_topmetr)),u1_struct_0(k3_topmetr)) )
=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(k3_topmetr))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k5_topmetr))
=> k1_binop_1(A,B,C) = k3_xcmplx_0(C,B) ) )
=> v5_pre_topc(A,k6_borsuk_1(k3_topmetr,k5_topmetr),k3_topmetr) ) ) ).
fof(d3_topalg_2,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
=> ( v2_topalg_2(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(k3_topmetr))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k3_topmetr))
=> ( ( r2_hidden(B,A)
& r2_hidden(C,A) )
=> r1_tarski(k1_rcomp_1(B,C),A) ) ) ) ) ) ).
fof(t8_topalg_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( v2_topalg_2(k1_rcomp_1(A,B))
& m1_subset_1(k1_rcomp_1(A,B),k1_zfmisc_1(u1_struct_0(k3_topmetr))) ) ) ) ).
fof(t9_topalg_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( v2_topalg_2(k2_rcomp_1(A,B))
& m1_subset_1(k2_rcomp_1(A,B),k1_zfmisc_1(u1_struct_0(k3_topmetr))) ) ) ) ).
fof(t10_topalg_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( v2_topalg_2(k1_rcomp_2(A,B))
& m1_subset_1(k1_rcomp_2(A,B),k1_zfmisc_1(u1_struct_0(k3_topmetr))) ) ) ) ).
fof(t11_topalg_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( v2_topalg_2(k2_rcomp_2(A,B))
& m1_subset_1(k2_rcomp_2(A,B),k1_zfmisc_1(u1_struct_0(k3_topmetr))) ) ) ) ).
fof(d4_topalg_2,axiom,
! [A] :
( m1_pre_topc(A,k3_topmetr)
=> ( v3_topalg_2(A)
<=> ( v2_topalg_2(k2_pre_topc(A))
& m1_subset_1(k2_pre_topc(A),k1_zfmisc_1(u1_struct_0(k3_topmetr))) ) ) ) ).
fof(t12_topalg_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_topalg_2(A)
& m1_pre_topc(A,k3_topalg_2) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> r1_tarski(k1_rcomp_1(B,C),u1_struct_0(A)) ) ) ) ).
fof(t13_topalg_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> v3_topalg_2(k4_topmetr(A,B)) ) ) ) ).
fof(t14_topalg_2,axiom,
v3_topalg_2(k5_topmetr) ).
fof(t15_topalg_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> v1_borsuk_2(k4_topmetr(A,B)) ) ) ) ).
fof(d5_topalg_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_topalg_2(A)
& m1_pre_topc(A,k3_topalg_2) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_borsuk_2(D,A,B,C)
=> ! [E] :
( m1_borsuk_2(E,A,B,C)
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(k6_borsuk_1(k5_topmetr,k5_topmetr)),u1_struct_0(A))
& m2_relset_1(F,u1_struct_0(k6_borsuk_1(k5_topmetr,k5_topmetr)),u1_struct_0(A)) )
=> ( F = k4_topalg_2(A,B,C,D,E)
<=> ! [G] :
( m1_subset_1(G,u1_struct_0(k5_topmetr))
=> ! [H] :
( m1_subset_1(H,u1_struct_0(k5_topmetr))
=> k1_binop_1(F,G,H) = k2_xcmplx_0(k3_xcmplx_0(k6_xcmplx_0(np__1,H),k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(A),D,G)),k3_xcmplx_0(H,k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(A),E,G))) ) ) ) ) ) ) ) ) ) ).
fof(t16_topalg_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_topalg_2(A)
& m1_pre_topc(A,k3_topalg_2) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_borsuk_2(D,A,B,C)
=> ! [E] :
( m1_borsuk_2(E,A,B,C)
=> r4_borsuk_2(A,B,C,D,E) ) ) ) ) ) ).
fof(t17_topalg_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_topalg_2(A)
& m1_pre_topc(A,k3_topalg_2) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_borsuk_2(C,A,B,B)
=> u1_struct_0(k3_topalg_1(A,B)) = k1_tarski(k6_eqrel_1(k1_topalg_1(A,B),k2_topalg_1(A,B),C)) ) ) ) ).
fof(t18_topalg_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k4_topmetr(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k4_topmetr(A,B)))
=> ! [E] :
( m1_borsuk_2(E,k4_topmetr(A,B),C,D)
=> ! [F] :
( m1_borsuk_2(F,k4_topmetr(A,B),C,D)
=> r3_borsuk_2(k4_topmetr(A,B),C,D,E,F) ) ) ) ) ) ) ) ).
fof(t19_topalg_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k4_topmetr(A,B)))
=> ! [D] :
( m1_borsuk_2(D,k4_topmetr(A,B),C,C)
=> u1_struct_0(k3_topalg_1(k4_topmetr(A,B),C)) = k1_tarski(k6_eqrel_1(k1_topalg_1(k4_topmetr(A,B),C),k2_topalg_1(k4_topmetr(A,B),C),D)) ) ) ) ) ) ).
fof(t20_topalg_2,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k5_topmetr))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k5_topmetr))
=> ! [C] :
( m1_borsuk_2(C,k5_topmetr,A,B)
=> ! [D] :
( m1_borsuk_2(D,k5_topmetr,A,B)
=> r3_borsuk_2(k5_topmetr,A,B,C,D) ) ) ) ) ).
fof(t21_topalg_2,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k5_topmetr))
=> ! [B] :
( m1_borsuk_2(B,k5_topmetr,A,A)
=> u1_struct_0(k3_topalg_1(k5_topmetr,A)) = k1_tarski(k6_eqrel_1(k1_topalg_1(k5_topmetr,A),k2_topalg_1(k5_topmetr,A),B)) ) ) ).
fof(dt_k1_topalg_2,axiom,
! [A,B,C,D,E,F] :
( ( m1_subset_1(A,k5_numbers)
& ~ v3_struct_0(B)
& v1_topalg_2(B,A)
& m1_pre_topc(B,k15_euclid(A))
& m1_subset_1(C,u1_struct_0(B))
& m1_subset_1(D,u1_struct_0(B))
& m1_borsuk_2(E,B,C,D)
& m1_borsuk_2(F,B,C,D) )
=> ( v1_funct_1(k1_topalg_2(A,B,C,D,E,F))
& v1_funct_2(k1_topalg_2(A,B,C,D,E,F),u1_struct_0(k6_borsuk_1(k5_topmetr,k5_topmetr)),u1_struct_0(B))
& m2_relset_1(k1_topalg_2(A,B,C,D,E,F),u1_struct_0(k6_borsuk_1(k5_topmetr,k5_topmetr)),u1_struct_0(B)) ) ) ).
fof(dt_k2_topalg_2,axiom,
! [A,B,C,D,E,F] :
( ( m1_subset_1(A,k5_numbers)
& ~ v3_struct_0(B)
& v1_topalg_2(B,A)
& m1_pre_topc(B,k15_euclid(A))
& m1_subset_1(C,u1_struct_0(B))
& m1_subset_1(D,u1_struct_0(B))
& m1_borsuk_2(E,B,C,D)
& m1_borsuk_2(F,B,C,D) )
=> m1_borsuk_6(k2_topalg_2(A,B,C,D,E,F),B,C,D,E,F) ) ).
fof(redefinition_k2_topalg_2,axiom,
! [A,B,C,D,E,F] :
( ( m1_subset_1(A,k5_numbers)
& ~ v3_struct_0(B)
& v1_topalg_2(B,A)
& m1_pre_topc(B,k15_euclid(A))
& m1_subset_1(C,u1_struct_0(B))
& m1_subset_1(D,u1_struct_0(B))
& m1_borsuk_2(E,B,C,D)
& m1_borsuk_2(F,B,C,D) )
=> k2_topalg_2(A,B,C,D,E,F) = k1_topalg_2(A,B,C,D,E,F) ) ).
fof(dt_k3_topalg_2,axiom,
( v1_pre_topc(k3_topalg_2)
& v3_topalg_2(k3_topalg_2)
& m1_pre_topc(k3_topalg_2,k3_topmetr) ) ).
fof(redefinition_k3_topalg_2,axiom,
k3_topalg_2 = k3_topmetr ).
fof(dt_k4_topalg_2,axiom,
! [A,B,C,D,E] :
( ( ~ v3_struct_0(A)
& v3_topalg_2(A)
& m1_pre_topc(A,k3_topalg_2)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A))
& m1_borsuk_2(D,A,B,C)
& m1_borsuk_2(E,A,B,C) )
=> ( v1_funct_1(k4_topalg_2(A,B,C,D,E))
& v1_funct_2(k4_topalg_2(A,B,C,D,E),u1_struct_0(k6_borsuk_1(k5_topmetr,k5_topmetr)),u1_struct_0(A))
& m2_relset_1(k4_topalg_2(A,B,C,D,E),u1_struct_0(k6_borsuk_1(k5_topmetr,k5_topmetr)),u1_struct_0(A)) ) ) ).
fof(dt_k5_topalg_2,axiom,
! [A,B,C,D,E] :
( ( ~ v3_struct_0(A)
& v3_topalg_2(A)
& m1_pre_topc(A,k3_topalg_2)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A))
& m1_borsuk_2(D,A,B,C)
& m1_borsuk_2(E,A,B,C) )
=> m1_borsuk_6(k5_topalg_2(A,B,C,D,E),A,B,C,D,E) ) ).
fof(redefinition_k5_topalg_2,axiom,
! [A,B,C,D,E] :
( ( ~ v3_struct_0(A)
& v3_topalg_2(A)
& m1_pre_topc(A,k3_topalg_2)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A))
& m1_borsuk_2(D,A,B,C)
& m1_borsuk_2(E,A,B,C) )
=> k5_topalg_2(A,B,C,D,E) = k4_topalg_2(A,B,C,D,E) ) ).
fof(t5_topalg_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> k1_rcomp_1(A,B) = a_2_0_topalg_2(A,B) ) ) ) ).
fof(fraenkel_a_2_0_topalg_2,axiom,
! [A,B,C] :
( ( v1_xreal_0(B)
& v1_xreal_0(C) )
=> ( r2_hidden(A,a_2_0_topalg_2(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_numbers)
& A = k2_xcmplx_0(k3_xcmplx_0(k5_real_1(np__1,D),B),k3_xcmplx_0(D,C))
& r1_xreal_0(np__0,D)
& r1_xreal_0(D,np__1) ) ) ) ).
%------------------------------------------------------------------------------