SET007 Axioms: SET007+838.ax


%------------------------------------------------------------------------------
% File     : SET007+838 : TPTP v7.5.0. 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 unit)
%            Number of atoms       :  324 (  20 equality)
%            Maximal formula depth :   26 (   9 average)
%            Number of connectives :  306 (  30 ~  ;   0  |; 146  &)
%                                         (   6 <=>; 124 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   35 (   0 propositional; 1-6 arity)
%            Number of functors    :   37 (   7 constant; 0-6 arity)
%            Number of variables   :  157 (   0 singleton; 152 !;   5 ?)
%            Maximal term depth    :    5 (   1 average)
% 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) ) ) ) )).
%------------------------------------------------------------------------------