SET007 Axioms: SET007+897.ax


%------------------------------------------------------------------------------
% File     : SET007+897 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Brouwer Fixed Point Theorem for Disks on the Plane
% 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    : brouwer [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   30 (   0 unit)
%            Number of atoms       :  292 (  41 equality)
%            Maximal formula depth :   26 (  13 average)
%            Number of connectives :  323 (  61 ~  ;   6  |; 141  &)
%                                         (   5 <=>; 110 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   22 (   0 propositional; 1-3 arity)
%            Number of functors    :   47 (   5 constant; 0-5 arity)
%            Number of variables   :  131 (   0 singleton; 124 !;   7 ?)
%            Maximal term depth    :   10 (   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(fc1_brouwer,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & ~ v3_realset2(A)
        & l1_pre_topc(A)
        & ~ v3_struct_0(B)
        & v2_pre_topc(B)
        & l1_pre_topc(B) )
     => ~ v1_xboole_0(k1_brouwer(A,B)) ) )).

fof(fc2_brouwer,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & ~ v3_struct_0(B)
        & v2_pre_topc(B)
        & ~ v3_realset2(B)
        & l1_pre_topc(B) )
     => ~ v1_xboole_0(k1_brouwer(A,B)) ) )).

fof(fc3_brouwer,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,u1_struct_0(k15_euclid(A)))
        & v1_xreal_0(C)
        & ~ v3_xreal_0(C) )
     => ( ~ v3_struct_0(k2_brouwer(A,B,C))
        & v2_pre_topc(k2_brouwer(A,B,C)) ) ) )).

fof(fc4_brouwer,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,u1_struct_0(k15_euclid(A)))
        & v1_xreal_0(C) )
     => ( v2_pre_topc(k2_brouwer(A,B,C))
        & v1_topalg_2(k2_brouwer(A,B,C),A) ) ) )).

fof(t1_brouwer,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( r2_hidden(C,k1_brouwer(A,B))
            <=> ? [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                  & ? [E] :
                      ( m1_subset_1(E,u1_struct_0(B))
                      & C = k8_borsuk_1(A,B,D,E)
                      & D != E ) ) ) ) ) )).

fof(t2_brouwer,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => k2_topreal9(A,B,np__0) = k1_struct_0(k15_euclid(A),B) ) ) )).

fof(d2_brouwer,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] :
              ( v1_xreal_0(C)
             => k2_brouwer(A,B,C) = k3_pre_topc(k15_euclid(A),k2_topreal9(A,B,C)) ) ) ) )).

fof(t3_brouwer,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => u1_struct_0(k2_brouwer(A,C,B)) = k2_topreal9(A,C,B) ) ) ) )).

fof(t4_brouwer,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_xreal_0(B)
            & ~ v3_xreal_0(B) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
                     => ~ ( C != D
                          & m1_subset_1(C,u1_struct_0(k2_brouwer(A,E,B)))
                          & ~ m1_subset_1(C,u1_struct_0(k7_toprealb(A,E,B)))
                          & ! [F] :
                              ( m1_subset_1(F,u1_struct_0(k7_toprealb(A,E,B)))
                             => k1_tarski(F) != k5_subset_1(u1_struct_0(k15_euclid(A)),k4_topreal9(A,C,D),k3_topreal9(A,E,B)) ) ) ) ) ) ) ) )).

fof(t5_brouwer,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_xreal_0(B)
            & ~ v3_xreal_0(B) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
                     => ~ ( C != D
                          & r2_hidden(C,u1_struct_0(k7_toprealb(A,E,B)))
                          & m1_subset_1(D,u1_struct_0(k2_brouwer(A,E,B)))
                          & ! [F] :
                              ( m1_subset_1(F,u1_struct_0(k7_toprealb(A,E,B)))
                             => ~ ( F != C
                                  & k2_tarski(C,F) = k5_subset_1(u1_struct_0(k15_euclid(A)),k4_topreal9(A,C,D),k3_topreal9(A,E,B)) ) ) ) ) ) ) ) ) )).

fof(d3_brouwer,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
                 => ! [E] :
                      ( ( v1_xreal_0(E)
                        & ~ v3_xreal_0(E) )
                     => ( ( m1_subset_1(C,u1_struct_0(k2_brouwer(A,B,E)))
                          & m1_subset_1(D,u1_struct_0(k2_brouwer(A,B,E))) )
                       => ( C = D
                          | ! [F] :
                              ( m1_subset_1(F,u1_struct_0(k15_euclid(A)))
                             => ( F = k3_brouwer(A,B,C,D,E)
                              <=> ( r2_hidden(F,k5_subset_1(u1_struct_0(k15_euclid(A)),k4_topreal9(A,C,D),k3_topreal9(A,B,E)))
                                  & F != C ) ) ) ) ) ) ) ) ) ) )).

fof(t6_brouwer,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & ~ v3_xreal_0(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(B)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(B)))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k15_euclid(B)))
                     => ( ( m1_subset_1(C,u1_struct_0(k2_brouwer(B,D,A)))
                          & m1_subset_1(E,u1_struct_0(k2_brouwer(B,D,A))) )
                       => ( C = E
                          | m1_subset_1(k3_brouwer(B,D,C,E,A),u1_struct_0(k7_toprealb(B,D,A))) ) ) ) ) ) ) ) )).

fof(t7_brouwer,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( v1_xreal_0(B)
            & ~ v3_xreal_0(B) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m2_subset_1(C,k1_numbers,k5_numbers) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(C)))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k15_euclid(C)))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(k15_euclid(C)))
                         => ! [G] :
                              ( m1_subset_1(G,k1_euclid(C))
                             => ! [H] :
                                  ( m1_subset_1(H,k1_euclid(C))
                                 => ! [I] :
                                      ( m1_subset_1(I,k1_euclid(C))
                                     => ( ( G = D
                                          & H = E
                                          & I = F
                                          & m1_subset_1(D,u1_struct_0(k2_brouwer(C,F,B)))
                                          & m1_subset_1(E,u1_struct_0(k2_brouwer(C,F,B)))
                                          & A = k7_xcmplx_0(k2_xcmplx_0(k4_xcmplx_0(k2_euclid_2(C,k20_euclid(C,E,D),k20_euclid(C,D,F))),k8_square_1(k6_xcmplx_0(k5_square_1(k2_euclid_2(C,k20_euclid(C,E,D),k20_euclid(C,D,F))),k3_xcmplx_0(k15_rvsum_1(k11_rvsum_1(k8_euclid(C,H,G))),k6_xcmplx_0(k15_rvsum_1(k11_rvsum_1(k8_euclid(C,G,I))),k5_square_1(B)))))),k15_rvsum_1(k11_rvsum_1(k8_euclid(C,H,G)))) )
                                       => ( D = E
                                          | k3_brouwer(C,F,D,E,B) = k17_euclid(C,k18_euclid(k6_xcmplx_0(np__1,A),C,D),k18_euclid(A,C,E)) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t8_brouwer,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( v1_xreal_0(B)
            & ~ v3_xreal_0(B) )
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ! [E] :
                      ( v1_xreal_0(E)
                     => ! [F] :
                          ( v1_xreal_0(F)
                         => ! [G] :
                              ( m1_subset_1(G,u1_struct_0(k15_euclid(np__2)))
                             => ! [H] :
                                  ( m1_subset_1(H,u1_struct_0(k15_euclid(np__2)))
                                 => ! [I] :
                                      ( m1_subset_1(I,u1_struct_0(k15_euclid(np__2)))
                                     => ( ( m1_subset_1(G,u1_struct_0(k2_brouwer(np__2,I,B)))
                                          & m1_subset_1(H,u1_struct_0(k2_brouwer(np__2,I,B)))
                                          & C = k5_real_1(k21_euclid(H),k21_euclid(G))
                                          & D = k5_real_1(k22_euclid(H),k22_euclid(G))
                                          & E = k5_real_1(k21_euclid(G),k21_euclid(I))
                                          & F = k5_real_1(k22_euclid(G),k22_euclid(I))
                                          & A = k7_xcmplx_0(k2_xcmplx_0(k4_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(E,C),k3_xcmplx_0(F,D))),k8_square_1(k6_xcmplx_0(k5_square_1(k2_xcmplx_0(k3_xcmplx_0(E,C),k3_xcmplx_0(F,D))),k3_xcmplx_0(k2_xcmplx_0(k5_square_1(C),k5_square_1(D)),k6_xcmplx_0(k2_xcmplx_0(k5_square_1(E),k5_square_1(F)),k5_square_1(B)))))),k2_xcmplx_0(k5_square_1(C),k5_square_1(D))) )
                                       => ( G = H
                                          | k3_brouwer(np__2,I,G,H,B) = k23_euclid(k2_xcmplx_0(k21_euclid(G),k3_xcmplx_0(A,C)),k2_xcmplx_0(k22_euclid(G),k3_xcmplx_0(A,D))) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d4_brouwer,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] :
              ( ( v1_xreal_0(C)
                & ~ v3_xreal_0(C) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k2_brouwer(A,B,C)))
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(k2_brouwer(A,B,C)),u1_struct_0(k2_brouwer(A,B,C)))
                        & m2_relset_1(E,u1_struct_0(k2_brouwer(A,B,C)),u1_struct_0(k2_brouwer(A,B,C))) )
                     => ( ~ r2_abian(u1_struct_0(k2_brouwer(A,B,C)),D,E)
                       => ! [F] :
                            ( m1_subset_1(F,u1_struct_0(k7_toprealb(A,B,C)))
                           => ( F = k4_brouwer(A,B,C,D,E)
                            <=> ? [G] :
                                  ( m1_subset_1(G,u1_struct_0(k15_euclid(A)))
                                  & ? [H] :
                                      ( m1_subset_1(H,u1_struct_0(k15_euclid(A)))
                                      & G = D
                                      & H = k8_funct_2(u1_struct_0(k2_brouwer(A,B,C)),u1_struct_0(k2_brouwer(A,B,C)),E,D)
                                      & F = k3_brouwer(A,B,H,G,C) ) ) ) ) ) ) ) ) ) ) )).

fof(t9_brouwer,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & ~ v3_xreal_0(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(B)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k2_brouwer(B,C,A)))
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(k2_brouwer(B,C,A)),u1_struct_0(k2_brouwer(B,C,A)))
                        & m2_relset_1(E,u1_struct_0(k2_brouwer(B,C,A)),u1_struct_0(k2_brouwer(B,C,A))) )
                     => ( m1_subset_1(D,u1_struct_0(k7_toprealb(B,C,A)))
                       => ( r2_abian(u1_struct_0(k2_brouwer(B,C,A)),D,E)
                          | k4_brouwer(B,C,A,D,E) = D ) ) ) ) ) ) ) )).

fof(t10_brouwer,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & m1_pre_topc(C,k2_brouwer(np__2,B,A)) )
             => ~ ( C = k7_toprealb(np__2,B,A)
                  & r1_borsuk_1(k2_brouwer(np__2,B,A),C) ) ) ) ) )).

fof(d5_brouwer,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( v1_xreal_0(B)
            & ~ v3_xreal_0(B) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(k2_brouwer(A,C,B)),u1_struct_0(k2_brouwer(A,C,B)))
                    & m2_relset_1(D,u1_struct_0(k2_brouwer(A,C,B)),u1_struct_0(k2_brouwer(A,C,B))) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(k2_brouwer(A,C,B)),u1_struct_0(k7_toprealb(A,C,B)))
                        & m2_relset_1(E,u1_struct_0(k2_brouwer(A,C,B)),u1_struct_0(k7_toprealb(A,C,B))) )
                     => ( E = k5_brouwer(A,B,C,D)
                      <=> ! [F] :
                            ( m1_subset_1(F,u1_struct_0(k2_brouwer(A,C,B)))
                           => k8_funct_2(u1_struct_0(k2_brouwer(A,C,B)),u1_struct_0(k7_toprealb(A,C,B)),E,F) = k4_brouwer(A,C,B,F,D) ) ) ) ) ) ) ) )).

fof(t11_brouwer,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & ~ v3_xreal_0(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(B)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k2_brouwer(B,C,A)))
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(k2_brouwer(B,C,A)),u1_struct_0(k2_brouwer(B,C,A)))
                        & m2_relset_1(E,u1_struct_0(k2_brouwer(B,C,A)),u1_struct_0(k2_brouwer(B,C,A))) )
                     => ( m1_subset_1(D,u1_struct_0(k7_toprealb(B,C,A)))
                       => ( r2_abian(u1_struct_0(k2_brouwer(B,C,A)),D,E)
                          | k8_funct_2(u1_struct_0(k2_brouwer(B,C,A)),u1_struct_0(k7_toprealb(B,C,A)),k5_brouwer(B,A,C,E),D) = D ) ) ) ) ) ) ) )).

fof(t12_brouwer,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & ~ v3_xreal_0(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(B)))
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(k2_brouwer(B,C,A)),u1_struct_0(k2_brouwer(B,C,A)))
                    & v5_pre_topc(D,k2_brouwer(B,C,A),k2_brouwer(B,C,A))
                    & m2_relset_1(D,u1_struct_0(k2_brouwer(B,C,A)),u1_struct_0(k2_brouwer(B,C,A))) )
                 => ( ~ r3_abian(D)
                   => k2_partfun1(u1_struct_0(k2_brouwer(B,C,A)),u1_struct_0(k7_toprealb(B,C,A)),k5_brouwer(B,A,C,D),k3_topreal9(B,C,A)) = k7_grcat_1(k7_toprealb(B,C,A)) ) ) ) ) ) )).

fof(t13_brouwer,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(k2_brouwer(np__2,B,A)),u1_struct_0(k2_brouwer(np__2,B,A)))
                & v5_pre_topc(C,k2_brouwer(np__2,B,A),k2_brouwer(np__2,B,A))
                & m2_relset_1(C,u1_struct_0(k2_brouwer(np__2,B,A)),u1_struct_0(k2_brouwer(np__2,B,A))) )
             => ( ~ r3_abian(C)
               => v5_pre_topc(k5_brouwer(np__2,A,B,C),k2_brouwer(np__2,B,A),k7_toprealb(np__2,B,A)) ) ) ) ) )).

fof(t14_brouwer,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & ~ v3_xreal_0(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(k2_brouwer(np__2,B,A)),u1_struct_0(k2_brouwer(np__2,B,A)))
                & v5_pre_topc(C,k2_brouwer(np__2,B,A),k2_brouwer(np__2,B,A))
                & m2_relset_1(C,u1_struct_0(k2_brouwer(np__2,B,A)),u1_struct_0(k2_brouwer(np__2,B,A))) )
             => r3_abian(C) ) ) ) )).

fof(t15_brouwer,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & ~ v3_xreal_0(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(k2_brouwer(np__2,B,A)),u1_struct_0(k2_brouwer(np__2,B,A)))
                & v5_pre_topc(C,k2_brouwer(np__2,B,A),k2_brouwer(np__2,B,A))
                & m2_relset_1(C,u1_struct_0(k2_brouwer(np__2,B,A)),u1_struct_0(k2_brouwer(np__2,B,A))) )
             => ? [D] :
                  ( m1_subset_1(D,u1_struct_0(k2_brouwer(np__2,B,A)))
                  & k8_funct_2(u1_struct_0(k2_brouwer(np__2,B,A)),u1_struct_0(k2_brouwer(np__2,B,A)),C,D) = D ) ) ) ) )).

fof(dt_k1_brouwer,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & ~ v3_struct_0(B)
        & v2_pre_topc(B)
        & l1_pre_topc(B) )
     => m1_subset_1(k1_brouwer(A,B),k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B)))) ) )).

fof(dt_k2_brouwer,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,u1_struct_0(k15_euclid(A)))
        & v1_xreal_0(C) )
     => m1_pre_topc(k2_brouwer(A,B,C),k15_euclid(A)) ) )).

fof(dt_k3_brouwer,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,u1_struct_0(k15_euclid(A)))
        & m1_subset_1(C,u1_struct_0(k15_euclid(A)))
        & m1_subset_1(D,u1_struct_0(k15_euclid(A)))
        & v1_xreal_0(E)
        & ~ v3_xreal_0(E) )
     => m1_subset_1(k3_brouwer(A,B,C,D,E),u1_struct_0(k15_euclid(A))) ) )).

fof(dt_k4_brouwer,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,u1_struct_0(k15_euclid(A)))
        & v1_xreal_0(C)
        & ~ v3_xreal_0(C)
        & m1_subset_1(D,u1_struct_0(k2_brouwer(A,B,C)))
        & v1_funct_1(E)
        & v1_funct_2(E,u1_struct_0(k2_brouwer(A,B,C)),u1_struct_0(k2_brouwer(A,B,C)))
        & m1_relset_1(E,u1_struct_0(k2_brouwer(A,B,C)),u1_struct_0(k2_brouwer(A,B,C))) )
     => m1_subset_1(k4_brouwer(A,B,C,D,E),u1_struct_0(k7_toprealb(A,B,C))) ) )).

fof(dt_k5_brouwer,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & v1_xreal_0(B)
        & ~ v3_xreal_0(B)
        & m1_subset_1(C,u1_struct_0(k15_euclid(A)))
        & v1_funct_1(D)
        & v1_funct_2(D,u1_struct_0(k2_brouwer(A,C,B)),u1_struct_0(k2_brouwer(A,C,B)))
        & m1_relset_1(D,u1_struct_0(k2_brouwer(A,C,B)),u1_struct_0(k2_brouwer(A,C,B))) )
     => ( v1_funct_1(k5_brouwer(A,B,C,D))
        & v1_funct_2(k5_brouwer(A,B,C,D),u1_struct_0(k2_brouwer(A,C,B)),u1_struct_0(k7_toprealb(A,C,B)))
        & m2_relset_1(k5_brouwer(A,B,C,D),u1_struct_0(k2_brouwer(A,C,B)),u1_struct_0(k7_toprealb(A,C,B))) ) ) )).

fof(d1_brouwer,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & l1_pre_topc(B) )
         => k1_brouwer(A,B) = a_2_0_brouwer(A,B) ) ) )).

fof(fraenkel_a_2_0_brouwer,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & v2_pre_topc(B)
        & l1_pre_topc(B)
        & ~ v3_struct_0(C)
        & v2_pre_topc(C)
        & l1_pre_topc(C) )
     => ( r2_hidden(A,a_2_0_brouwer(B,C))
      <=> ? [D,E] :
            ( m1_subset_1(D,u1_struct_0(B))
            & m1_subset_1(E,u1_struct_0(C))
            & A = k8_borsuk_1(B,C,D,E)
            & D != E ) ) ) )).
%------------------------------------------------------------------------------