SET007 Axioms: SET007+346.ax


%------------------------------------------------------------------------------
% File     : SET007+346 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Brouwer Fixed Point Theorem for Intervals
% 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    : treal_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   35 (   3 unit)
%            Number of atoms       :  257 (  47 equality)
%            Maximal formula depth :   24 (  10 average)
%            Number of connectives :  238 (  16 ~  ;   0  |;  84  &)
%                                         (   2 <=>; 136 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   15 (   0 propositional; 1-3 arity)
%            Number of functors    :   29 (   7 constant; 0-5 arity)
%            Number of variables   :  122 (   0 singleton; 116 !;   6 ?)
%            Maximal term depth    :    5 (   2 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(t1_treal_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( ( r1_xreal_0(A,B)
                      & r1_xreal_0(C,D) )
                   => r1_tarski(k1_rcomp_1(B,C),k1_rcomp_1(A,D)) ) ) ) ) ) )).

fof(t2_treal_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( ( r1_xreal_0(A,B)
                      & r1_xreal_0(C,D)
                      & r1_xreal_0(B,C) )
                   => k4_subset_1(k1_numbers,k1_rcomp_1(A,C),k1_rcomp_1(B,D)) = k1_rcomp_1(A,D) ) ) ) ) ) )).

fof(t3_treal_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( ( r1_xreal_0(A,B)
                      & r1_xreal_0(C,D)
                      & r1_xreal_0(B,C) )
                   => k5_subset_1(k1_numbers,k1_rcomp_1(A,C),k1_rcomp_1(B,D)) = k1_rcomp_1(B,C) ) ) ) ) ) )).

fof(t4_treal_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
             => ( C = k1_rcomp_1(A,B)
               => v4_pre_topc(C,k3_topmetr) ) ) ) ) )).

fof(t5_treal_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
           => ( v1_borsuk_1(k4_topmetr(A,B),k3_topmetr)
              & m1_pre_topc(k4_topmetr(A,B),k3_topmetr) ) ) ) ) )).

fof(t6_treal_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( ( r1_xreal_0(A,B)
                      & r1_xreal_0(C,D)
                      & r1_xreal_0(B,C) )
                   => ( v1_borsuk_1(k4_topmetr(B,C),k4_topmetr(A,D))
                      & m1_pre_topc(k4_topmetr(B,C),k4_topmetr(A,D)) ) ) ) ) ) ) )).

fof(t7_treal_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( ( r1_xreal_0(A,B)
                      & r1_xreal_0(C,D)
                      & r1_xreal_0(B,C) )
                   => ( k4_topmetr(A,D) = k1_tsep_1(k3_topmetr,k4_topmetr(A,C),k4_topmetr(B,D))
                      & k4_topmetr(B,C) = k2_tsep_1(k3_topmetr,k4_topmetr(A,C),k4_topmetr(B,D)) ) ) ) ) ) ) )).

fof(d1_treal_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
           => k1_treal_1(A,B) = A ) ) ) )).

fof(d2_treal_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
           => k2_treal_1(A,B) = B ) ) ) )).

fof(t8_treal_1,axiom,
    ( k23_borsuk_1 = k1_treal_1(np__0,np__1)
    & k24_borsuk_1 = k2_treal_1(np__0,np__1) )).

fof(t9_treal_1,axiom,(
    ! [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) )
               => ( k1_treal_1(A,B) = k1_treal_1(A,C)
                  & k2_treal_1(B,C) = k2_treal_1(A,C) ) ) ) ) ) )).

fof(d3_treal_1,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] :
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,u1_struct_0(k4_topmetr(np__0,np__1)),u1_struct_0(k4_topmetr(A,B)))
                          & m2_relset_1(E,u1_struct_0(k4_topmetr(np__0,np__1)),u1_struct_0(k4_topmetr(A,B))) )
                       => ( E = k3_treal_1(A,B,C,D)
                        <=> ! [F] :
                              ( m1_subset_1(F,u1_struct_0(k4_topmetr(np__0,np__1)))
                             => ! [G] :
                                  ( v1_xreal_0(G)
                                 => ! [H] :
                                      ( v1_xreal_0(H)
                                     => ! [I] :
                                          ( v1_xreal_0(I)
                                         => ( ( F = G
                                              & H = C
                                              & I = D )
                                           => k8_funct_2(u1_struct_0(k4_topmetr(np__0,np__1)),u1_struct_0(k4_topmetr(A,B)),E,F) = k2_xcmplx_0(k3_xcmplx_0(k6_xcmplx_0(np__1,G),H),k3_xcmplx_0(G,I)) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t10_treal_1,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_subset_1(E,u1_struct_0(k4_topmetr(np__0,np__1)))
                       => ! [F] :
                            ( v1_xreal_0(F)
                           => ! [G] :
                                ( v1_xreal_0(G)
                               => ! [H] :
                                    ( v1_xreal_0(H)
                                   => ( ( E = F
                                        & G = C
                                        & H = D )
                                     => k8_funct_2(u1_struct_0(k4_topmetr(np__0,np__1)),u1_struct_0(k4_topmetr(A,B)),k3_treal_1(A,B,C,D),E) = k2_xcmplx_0(k3_xcmplx_0(k6_xcmplx_0(H,G),F),G) ) ) ) ) ) ) ) ) ) ) )).

fof(t11_treal_1,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)))
                   => ( v1_funct_1(k3_treal_1(A,B,C,D))
                      & v1_funct_2(k3_treal_1(A,B,C,D),u1_struct_0(k4_topmetr(np__0,np__1)),u1_struct_0(k4_topmetr(A,B)))
                      & v5_pre_topc(k3_treal_1(A,B,C,D),k4_topmetr(np__0,np__1),k4_topmetr(A,B))
                      & m2_relset_1(k3_treal_1(A,B,C,D),u1_struct_0(k4_topmetr(np__0,np__1)),u1_struct_0(k4_topmetr(A,B))) ) ) ) ) ) ) )).

fof(t12_treal_1,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)))
                   => ( k8_funct_2(u1_struct_0(k4_topmetr(np__0,np__1)),u1_struct_0(k4_topmetr(A,B)),k3_treal_1(A,B,C,D),k1_treal_1(np__0,np__1)) = C
                      & k8_funct_2(u1_struct_0(k4_topmetr(np__0,np__1)),u1_struct_0(k4_topmetr(A,B)),k3_treal_1(A,B,C,D),k2_treal_1(np__0,np__1)) = D ) ) ) ) ) ) )).

fof(t13_treal_1,axiom,(
    k3_treal_1(np__0,np__1,k1_treal_1(np__0,np__1),k2_treal_1(np__0,np__1)) = k7_grcat_1(k4_topmetr(np__0,np__1)) )).

fof(d4_treal_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ~ r1_xreal_0(B,A)
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(k4_topmetr(np__0,np__1)))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(k4_topmetr(np__0,np__1)))
                   => ! [E] :
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(np__0,np__1)))
                          & m2_relset_1(E,u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(np__0,np__1))) )
                       => ( E = k4_treal_1(A,B,C,D)
                        <=> ! [F] :
                              ( m1_subset_1(F,u1_struct_0(k4_topmetr(A,B)))
                             => ! [G] :
                                  ( v1_xreal_0(G)
                                 => ! [H] :
                                      ( v1_xreal_0(H)
                                     => ! [I] :
                                          ( v1_xreal_0(I)
                                         => ( ( F = G
                                              & H = C
                                              & I = D )
                                           => k8_funct_2(u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(np__0,np__1)),E,F) = k7_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(k6_xcmplx_0(B,G),H),k3_xcmplx_0(k6_xcmplx_0(G,A),I)),k6_xcmplx_0(B,A)) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t14_treal_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ~ r1_xreal_0(B,A)
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(k4_topmetr(np__0,np__1)))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(k4_topmetr(np__0,np__1)))
                   => ! [E] :
                        ( m1_subset_1(E,u1_struct_0(k4_topmetr(A,B)))
                       => ! [F] :
                            ( m1_subset_1(F,k1_numbers)
                           => ! [G] :
                                ( m1_subset_1(G,k1_numbers)
                               => ! [H] :
                                    ( m1_subset_1(H,k1_numbers)
                                   => ( ( E = F
                                        & G = C
                                        & H = D )
                                     => k8_funct_2(u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(np__0,np__1)),k4_treal_1(A,B,C,D),E) = k2_xcmplx_0(k3_xcmplx_0(k7_xcmplx_0(k5_real_1(H,G),k6_xcmplx_0(B,A)),F),k7_xcmplx_0(k6_xcmplx_0(k3_xcmplx_0(B,G),k3_xcmplx_0(A,H)),k6_xcmplx_0(B,A))) ) ) ) ) ) ) ) ) ) ) )).

fof(t15_treal_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ~ r1_xreal_0(B,A)
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(k4_topmetr(np__0,np__1)))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(k4_topmetr(np__0,np__1)))
                   => ( v1_funct_1(k4_treal_1(A,B,C,D))
                      & v1_funct_2(k4_treal_1(A,B,C,D),u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(np__0,np__1)))
                      & v5_pre_topc(k4_treal_1(A,B,C,D),k4_topmetr(A,B),k4_topmetr(np__0,np__1))
                      & m2_relset_1(k4_treal_1(A,B,C,D),u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(np__0,np__1))) ) ) ) ) ) ) )).

fof(t16_treal_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ~ r1_xreal_0(B,A)
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(k4_topmetr(np__0,np__1)))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(k4_topmetr(np__0,np__1)))
                   => ( k8_funct_2(u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(np__0,np__1)),k4_treal_1(A,B,C,D),k1_treal_1(A,B)) = C
                      & k8_funct_2(u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(np__0,np__1)),k4_treal_1(A,B,C,D),k2_treal_1(A,B)) = D ) ) ) ) ) ) )).

fof(t17_treal_1,axiom,(
    k4_treal_1(np__0,np__1,k1_treal_1(np__0,np__1),k2_treal_1(np__0,np__1)) = k7_grcat_1(k4_topmetr(np__0,np__1)) )).

fof(t18_treal_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ~ r1_xreal_0(B,A)
           => ( k7_grcat_1(k4_topmetr(A,B)) = k7_funct_2(u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(np__0,np__1)),u1_struct_0(k4_topmetr(A,B)),k4_treal_1(A,B,k1_treal_1(np__0,np__1),k2_treal_1(np__0,np__1)),k3_treal_1(A,B,k1_treal_1(A,B),k2_treal_1(A,B)))
              & k7_grcat_1(k4_topmetr(np__0,np__1)) = k7_funct_2(u1_struct_0(k4_topmetr(np__0,np__1)),u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(np__0,np__1)),k3_treal_1(A,B,k1_treal_1(A,B),k2_treal_1(A,B)),k4_treal_1(A,B,k1_treal_1(np__0,np__1),k2_treal_1(np__0,np__1))) ) ) ) ) )).

fof(t19_treal_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ~ r1_xreal_0(B,A)
           => ( k7_grcat_1(k4_topmetr(A,B)) = k7_funct_2(u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(np__0,np__1)),u1_struct_0(k4_topmetr(A,B)),k4_treal_1(A,B,k2_treal_1(np__0,np__1),k1_treal_1(np__0,np__1)),k3_treal_1(A,B,k2_treal_1(A,B),k1_treal_1(A,B)))
              & k7_grcat_1(k4_topmetr(np__0,np__1)) = k7_funct_2(u1_struct_0(k4_topmetr(np__0,np__1)),u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(np__0,np__1)),k3_treal_1(A,B,k2_treal_1(A,B),k1_treal_1(A,B)),k4_treal_1(A,B,k2_treal_1(np__0,np__1),k1_treal_1(np__0,np__1))) ) ) ) ) )).

fof(t20_treal_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ~ r1_xreal_0(B,A)
           => ( v3_tops_2(k3_treal_1(A,B,k1_treal_1(A,B),k2_treal_1(A,B)),k4_topmetr(np__0,np__1),k4_topmetr(A,B))
              & k2_tops_2(k4_topmetr(np__0,np__1),k4_topmetr(A,B),k3_treal_1(A,B,k1_treal_1(A,B),k2_treal_1(A,B))) = k4_treal_1(A,B,k1_treal_1(np__0,np__1),k2_treal_1(np__0,np__1))
              & v3_tops_2(k4_treal_1(A,B,k1_treal_1(np__0,np__1),k2_treal_1(np__0,np__1)),k4_topmetr(A,B),k4_topmetr(np__0,np__1))
              & k2_tops_2(k4_topmetr(A,B),k4_topmetr(np__0,np__1),k4_treal_1(A,B,k1_treal_1(np__0,np__1),k2_treal_1(np__0,np__1))) = k3_treal_1(A,B,k1_treal_1(A,B),k2_treal_1(A,B)) ) ) ) ) )).

fof(t21_treal_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ~ r1_xreal_0(B,A)
           => ( v3_tops_2(k3_treal_1(A,B,k2_treal_1(A,B),k1_treal_1(A,B)),k4_topmetr(np__0,np__1),k4_topmetr(A,B))
              & k2_tops_2(k4_topmetr(np__0,np__1),k4_topmetr(A,B),k3_treal_1(A,B,k2_treal_1(A,B),k1_treal_1(A,B))) = k4_treal_1(A,B,k2_treal_1(np__0,np__1),k1_treal_1(np__0,np__1))
              & v3_tops_2(k4_treal_1(A,B,k2_treal_1(np__0,np__1),k1_treal_1(np__0,np__1)),k4_topmetr(A,B),k4_topmetr(np__0,np__1))
              & k2_tops_2(k4_topmetr(A,B),k4_topmetr(np__0,np__1),k4_treal_1(A,B,k2_treal_1(np__0,np__1),k1_treal_1(np__0,np__1))) = k3_treal_1(A,B,k2_treal_1(A,B),k1_treal_1(A,B)) ) ) ) ) )).

fof(t22_treal_1,axiom,(
    v1_connsp_1(k22_borsuk_1) )).

fof(t23_treal_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
           => v1_connsp_1(k4_topmetr(A,B)) ) ) ) )).

fof(t24_treal_1,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,u1_struct_0(k22_borsuk_1),u1_struct_0(k22_borsuk_1))
        & v5_pre_topc(A,k22_borsuk_1,k22_borsuk_1)
        & m2_relset_1(A,u1_struct_0(k22_borsuk_1),u1_struct_0(k22_borsuk_1)) )
     => ? [B] :
          ( m1_subset_1(B,u1_struct_0(k22_borsuk_1))
          & k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k22_borsuk_1),A,B) = B ) ) )).

fof(t25_treal_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
           => ! [C] :
                ( ( v1_funct_1(C)
                  & v1_funct_2(C,u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(A,B)))
                  & v5_pre_topc(C,k4_topmetr(A,B),k4_topmetr(A,B))
                  & m2_relset_1(C,u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(A,B))) )
               => ? [D] :
                    ( m1_subset_1(D,u1_struct_0(k4_topmetr(A,B)))
                    & k8_funct_2(u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(A,B)),C,D) = D ) ) ) ) ) )).

fof(t26_treal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & m1_pre_topc(A,k3_topmetr) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & m1_pre_topc(B,k3_topmetr) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & v5_pre_topc(C,A,B)
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ~ ( ? [D] :
                      ( m1_subset_1(D,k1_numbers)
                      & ? [E] :
                          ( m1_subset_1(E,k1_numbers)
                          & r1_xreal_0(D,E)
                          & r1_tarski(k1_rcomp_1(D,E),u1_struct_0(A))
                          & r1_tarski(k1_rcomp_1(D,E),u1_struct_0(B))
                          & r1_tarski(k4_pre_topc(A,B,C,k1_rcomp_1(D,E)),k1_rcomp_1(D,E)) ) )
                  & ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D) != D ) ) ) ) ) )).

fof(t27_treal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & m1_pre_topc(A,k3_topmetr) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & m1_pre_topc(B,k3_topmetr) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & v5_pre_topc(C,A,B)
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ~ ( ? [D] :
                      ( m1_subset_1(D,k1_numbers)
                      & ? [E] :
                          ( m1_subset_1(E,k1_numbers)
                          & r1_xreal_0(D,E)
                          & r1_tarski(k1_rcomp_1(D,E),u1_struct_0(A))
                          & r1_tarski(k4_pre_topc(A,B,C,k1_rcomp_1(D,E)),k1_rcomp_1(D,E)) ) )
                  & ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D) != D ) ) ) ) ) )).

fof(dt_k1_treal_1,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => m1_subset_1(k1_treal_1(A,B),u1_struct_0(k4_topmetr(A,B))) ) )).

fof(dt_k2_treal_1,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => m1_subset_1(k2_treal_1(A,B),u1_struct_0(k4_topmetr(A,B))) ) )).

fof(dt_k3_treal_1,axiom,(
    ! [A,B,C,D] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B)
        & m1_subset_1(C,u1_struct_0(k4_topmetr(A,B)))
        & m1_subset_1(D,u1_struct_0(k4_topmetr(A,B))) )
     => ( v1_funct_1(k3_treal_1(A,B,C,D))
        & v1_funct_2(k3_treal_1(A,B,C,D),u1_struct_0(k4_topmetr(np__0,np__1)),u1_struct_0(k4_topmetr(A,B)))
        & m2_relset_1(k3_treal_1(A,B,C,D),u1_struct_0(k4_topmetr(np__0,np__1)),u1_struct_0(k4_topmetr(A,B))) ) ) )).

fof(dt_k4_treal_1,axiom,(
    ! [A,B,C,D] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B)
        & m1_subset_1(C,u1_struct_0(k4_topmetr(np__0,np__1)))
        & m1_subset_1(D,u1_struct_0(k4_topmetr(np__0,np__1))) )
     => ( v1_funct_1(k4_treal_1(A,B,C,D))
        & v1_funct_2(k4_treal_1(A,B,C,D),u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(np__0,np__1)))
        & m2_relset_1(k4_treal_1(A,B,C,D),u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(np__0,np__1))) ) ) )).
%------------------------------------------------------------------------------