SET007 Axioms: SET007+785.ax


%------------------------------------------------------------------------------
% File     : SET007+785 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Lines in n-Dimensional Euclidean Spaces
% 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    : euclid_4 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   89 (   0 unit)
%            Number of atoms       :  448 (  81 equality)
%            Maximal formula depth :   20 (   9 average)
%            Number of connectives :  391 (  32 ~  ;   2  |; 100  &)
%                                         (  10 <=>; 247 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   13 (   0 propositional; 1-3 arity)
%            Number of functors    :   45 (   6 constant; 0-4 arity)
%            Number of variables   :  299 (   0 singleton; 291 !;   8 ?)
%            Maximal term depth    :    6 (   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_euclid_4,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k1_euclid(A))
        & m1_subset_1(C,k1_euclid(A)) )
     => ~ v1_xboole_0(k1_euclid_4(A,B,C)) ) )).

fof(fc2_euclid_4,axiom,(
    ! [A,B,C] :
      ( ( 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))) )
     => ~ v1_xboole_0(k6_euclid_4(A,B,C)) ) )).

fof(t1_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ( k7_euclid(A,k9_euclid(A,np__0,B),B) = B
            & k7_euclid(A,B,k5_euclid(A)) = B ) ) ) )).

fof(t2_euclid_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k9_euclid(B,A,k5_euclid(B)) = k5_euclid(B) ) ) )).

fof(t3_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ( k9_euclid(A,np__1,B) = B
            & k9_euclid(A,np__0,B) = k5_euclid(A) ) ) ) )).

fof(t4_euclid_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_euclid(C))
                 => k9_euclid(C,k4_real_1(A,B),D) = k9_euclid(C,A,k9_euclid(C,B,D)) ) ) ) ) )).

fof(t5_euclid_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(B))
             => ~ ( k9_euclid(B,A,C) = k5_euclid(B)
                  & A != np__0
                  & C != k5_euclid(B) ) ) ) ) )).

fof(t6_euclid_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(B))
             => ! [D] :
                  ( m1_subset_1(D,k1_euclid(B))
                 => k9_euclid(B,A,k7_euclid(B,C,D)) = k7_euclid(B,k9_euclid(B,A,C),k9_euclid(B,A,D)) ) ) ) ) )).

fof(t7_euclid_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_euclid(C))
                 => k9_euclid(C,k3_real_1(A,B),D) = k7_euclid(C,k9_euclid(C,A,D),k9_euclid(C,B,D)) ) ) ) ) )).

fof(t8_euclid_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(B))
             => ! [D] :
                  ( m1_subset_1(D,k1_euclid(B))
                 => ~ ( k9_euclid(B,A,C) = k9_euclid(B,A,D)
                      & A != np__0
                      & C != D ) ) ) ) ) )).

fof(t9_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(A))
             => k1_euclid_4(A,B,C) = k1_euclid_4(A,C,B) ) ) ) )).

fof(t10_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(A))
             => ( r2_hidden(B,k2_euclid_4(A,B,C))
                & r2_hidden(C,k2_euclid_4(A,B,C)) ) ) ) ) )).

fof(t11_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_euclid(A))
                 => ! [E] :
                      ( m1_subset_1(E,k1_euclid(A))
                     => ( ( r2_hidden(B,k2_euclid_4(A,C,D))
                          & r2_hidden(E,k2_euclid_4(A,C,D)) )
                       => r1_tarski(k2_euclid_4(A,B,E),k2_euclid_4(A,C,D)) ) ) ) ) ) ) )).

fof(t12_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_euclid(A))
                 => ! [E] :
                      ( m1_subset_1(E,k1_euclid(A))
                     => ( ( r2_hidden(B,k2_euclid_4(A,C,D))
                          & r2_hidden(E,k2_euclid_4(A,C,D)) )
                       => ( B = E
                          | r1_tarski(k2_euclid_4(A,C,D),k2_euclid_4(A,B,E)) ) ) ) ) ) ) ) )).

fof(d2_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_euclid(A)))
         => ( v1_euclid_4(B,A)
          <=> ? [C] :
                ( m1_subset_1(C,k1_euclid(A))
                & ? [D] :
                    ( m1_subset_1(D,k1_euclid(A))
                    & C != D
                    & B = k2_euclid_4(A,C,D) ) ) ) ) ) )).

fof(t13_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_euclid(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_euclid(A)))
             => ! [D] :
                  ( m1_subset_1(D,k1_euclid(A))
                 => ! [E] :
                      ( m1_subset_1(E,k1_euclid(A))
                     => ~ ( v1_euclid_4(B,A)
                          & v1_euclid_4(C,A)
                          & r2_hidden(D,B)
                          & r2_hidden(E,B)
                          & r2_hidden(D,C)
                          & r2_hidden(E,C)
                          & D != E
                          & B != C ) ) ) ) ) ) )).

fof(t14_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_euclid(A)))
         => ~ ( v1_euclid_4(B,A)
              & ! [C] :
                  ( m1_subset_1(C,k1_euclid(A))
                 => ! [D] :
                      ( m1_subset_1(D,k1_euclid(A))
                     => ~ ( r2_hidden(C,B)
                          & r2_hidden(D,B)
                          & C != D ) ) ) ) ) ) )).

fof(t15_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_euclid(A)))
             => ~ ( v1_euclid_4(C,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_euclid(A))
                     => ~ ( B != D
                          & r2_hidden(D,C) ) ) ) ) ) ) )).

fof(d3_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => k3_euclid_4(A,B) = B ) ) )).

fof(d4_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => k4_euclid_4(A,B) = k12_euclid(k3_euclid_4(A,B)) ) ) )).

fof(d5_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(A))
             => k5_euclid_4(A,B,C) = k1_euclid_2(k3_euclid_4(A,B),k3_euclid_4(A,C)) ) ) ) )).

fof(t16_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(A))
             => k5_euclid_4(A,B,C) = k4_real_1(k6_real_1(np__1,np__4),k5_real_1(k7_square_1(k4_euclid_4(A,k7_euclid(A,B,C))),k7_square_1(k4_euclid_4(A,k8_euclid(A,B,C))))) ) ) ) )).

fof(t17_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => r1_xreal_0(np__0,k5_euclid_4(A,B,B)) ) ) )).

fof(t18_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => k7_square_1(k4_euclid_4(A,B)) = k5_euclid_4(A,B,B) ) ) )).

fof(t19_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => r1_xreal_0(np__0,k4_euclid_4(A,B)) ) ) )).

fof(t20_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => k4_euclid_4(A,B) = k9_square_1(k5_euclid_4(A,B,B)) ) ) )).

fof(t21_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ( k5_euclid_4(A,B,B) = np__0
          <=> k4_euclid_4(A,B) = np__0 ) ) ) )).

fof(t22_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ( k5_euclid_4(A,B,B) = np__0
          <=> B = k5_euclid(A) ) ) ) )).

fof(t23_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => k5_euclid_4(A,B,k5_euclid(A)) = np__0 ) ) )).

fof(t24_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => k5_euclid_4(A,k5_euclid(A),B) = np__0 ) ) )).

fof(t25_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_euclid(A))
                 => k5_euclid_4(A,k7_euclid(A,B,C),D) = k3_real_1(k5_euclid_4(A,B,D),k5_euclid_4(A,C,D)) ) ) ) ) )).

fof(t26_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(A))
             => ! [D] :
                  ( v1_xreal_0(D)
                 => k5_euclid_4(A,k9_euclid(A,D,B),C) = k3_xcmplx_0(D,k5_euclid_4(A,B,C)) ) ) ) ) )).

fof(t27_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(A))
             => ! [D] :
                  ( v1_xreal_0(D)
                 => k5_euclid_4(A,B,k9_euclid(A,D,C)) = k3_xcmplx_0(D,k5_euclid_4(A,B,C)) ) ) ) ) )).

fof(t28_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(A))
             => k5_euclid_4(A,k6_euclid(A,B),C) = k1_real_1(k5_euclid_4(A,B,C)) ) ) ) )).

fof(t29_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(A))
             => k5_euclid_4(A,B,k6_euclid(A,C)) = k1_real_1(k5_euclid_4(A,B,C)) ) ) ) )).

fof(t30_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(A))
             => k5_euclid_4(A,k6_euclid(A,B),k6_euclid(A,C)) = k5_euclid_4(A,B,C) ) ) ) )).

fof(t31_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_euclid(A))
                 => k5_euclid_4(A,k8_euclid(A,B,C),D) = k5_real_1(k5_euclid_4(A,B,D),k5_euclid_4(A,C,D)) ) ) ) ) )).

fof(t32_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( m1_subset_1(D,k1_euclid(A))
                 => ! [E] :
                      ( m1_subset_1(E,k1_euclid(A))
                     => ! [F] :
                          ( m1_subset_1(F,k1_euclid(A))
                         => k5_euclid_4(A,k7_euclid(A,k9_euclid(A,B,D),k9_euclid(A,C,E)),F) = k2_xcmplx_0(k3_xcmplx_0(B,k5_euclid_4(A,D,F)),k3_xcmplx_0(C,k5_euclid_4(A,E,F))) ) ) ) ) ) ) )).

fof(t33_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_euclid(A))
                 => k5_euclid_4(A,B,k7_euclid(A,C,D)) = k3_real_1(k5_euclid_4(A,B,C),k5_euclid_4(A,B,D)) ) ) ) ) )).

fof(t34_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_euclid(A))
                 => k5_euclid_4(A,B,k8_euclid(A,C,D)) = k5_real_1(k5_euclid_4(A,B,C),k5_euclid_4(A,B,D)) ) ) ) ) )).

fof(t35_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_euclid(A))
                 => ! [E] :
                      ( m1_subset_1(E,k1_euclid(A))
                     => k5_euclid_4(A,k7_euclid(A,B,C),k7_euclid(A,D,E)) = k3_real_1(k3_real_1(k3_real_1(k5_euclid_4(A,B,D),k5_euclid_4(A,B,E)),k5_euclid_4(A,C,D)),k5_euclid_4(A,C,E)) ) ) ) ) ) )).

fof(t36_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_euclid(A))
                 => ! [E] :
                      ( m1_subset_1(E,k1_euclid(A))
                     => k5_euclid_4(A,k8_euclid(A,B,C),k8_euclid(A,D,E)) = k3_real_1(k5_real_1(k5_real_1(k5_euclid_4(A,B,D),k5_euclid_4(A,B,E)),k5_euclid_4(A,C,D)),k5_euclid_4(A,C,E)) ) ) ) ) ) )).

fof(t37_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(A))
             => k5_euclid_4(A,k7_euclid(A,B,C),k7_euclid(A,B,C)) = k3_real_1(k3_real_1(k5_euclid_4(A,B,B),k4_real_1(np__2,k5_euclid_4(A,B,C))),k5_euclid_4(A,C,C)) ) ) ) )).

fof(t38_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(A))
             => k5_euclid_4(A,k8_euclid(A,B,C),k8_euclid(A,B,C)) = k3_real_1(k5_real_1(k5_euclid_4(A,B,B),k4_real_1(np__2,k5_euclid_4(A,B,C))),k5_euclid_4(A,C,C)) ) ) ) )).

fof(t39_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(A))
             => k7_square_1(k4_euclid_4(A,k7_euclid(A,B,C))) = k3_real_1(k3_real_1(k7_square_1(k4_euclid_4(A,B)),k4_real_1(np__2,k5_euclid_4(A,B,C))),k7_square_1(k4_euclid_4(A,C))) ) ) ) )).

fof(t40_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(A))
             => k7_square_1(k4_euclid_4(A,k8_euclid(A,B,C))) = k3_real_1(k5_real_1(k7_square_1(k4_euclid_4(A,B)),k4_real_1(np__2,k5_euclid_4(A,B,C))),k7_square_1(k4_euclid_4(A,C))) ) ) ) )).

fof(t41_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(A))
             => k3_real_1(k7_square_1(k4_euclid_4(A,k7_euclid(A,B,C))),k7_square_1(k4_euclid_4(A,k8_euclid(A,B,C)))) = k4_real_1(np__2,k3_real_1(k7_square_1(k4_euclid_4(A,B)),k7_square_1(k4_euclid_4(A,C)))) ) ) ) )).

fof(t42_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(A))
             => k5_real_1(k7_square_1(k4_euclid_4(A,k7_euclid(A,B,C))),k7_square_1(k4_euclid_4(A,k8_euclid(A,B,C)))) = k4_real_1(np__4,k5_euclid_4(A,B,C)) ) ) ) )).

fof(t43_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(A))
             => r1_xreal_0(k18_complex1(k5_euclid_4(A,B,C)),k4_real_1(k4_euclid_4(A,B),k4_euclid_4(A,C))) ) ) ) )).

fof(t44_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(A))
             => r1_xreal_0(k4_euclid_4(A,k7_euclid(A,B,C)),k3_real_1(k4_euclid_4(A,B),k4_euclid_4(A,C))) ) ) ) )).

fof(d6_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(A))
             => ( r1_euclid_4(A,B,C)
              <=> k5_euclid_4(A,B,C) = np__0 ) ) ) ) )).

fof(t46_euclid_4,axiom,(
    ! [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)))
             => k6_euclid_4(A,B,C) = k6_euclid_4(A,C,B) ) ) ) )).

fof(t47_euclid_4,axiom,(
    ! [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)))
             => ( r2_hidden(B,k7_euclid_4(A,B,C))
                & r2_hidden(C,k7_euclid_4(A,B,C)) ) ) ) ) )).

fof(t48_euclid_4,axiom,(
    ! [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] :
                      ( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
                     => ( ( r2_hidden(B,k7_euclid_4(A,C,D))
                          & r2_hidden(E,k7_euclid_4(A,C,D)) )
                       => r1_tarski(k7_euclid_4(A,B,E),k7_euclid_4(A,C,D)) ) ) ) ) ) ) )).

fof(t49_euclid_4,axiom,(
    ! [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] :
                      ( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
                     => ( ( r2_hidden(B,k7_euclid_4(A,C,D))
                          & r2_hidden(E,k7_euclid_4(A,C,D)) )
                       => ( B = E
                          | r1_tarski(k7_euclid_4(A,C,D),k7_euclid_4(A,B,E)) ) ) ) ) ) ) ) )).

fof(d8_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
         => ( v2_euclid_4(B,A)
          <=> ? [C] :
                ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
                & ? [D] :
                    ( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
                    & C != D
                    & B = k7_euclid_4(A,C,D) ) ) ) ) ) )).

fof(t50_euclid_4,axiom,(
    ! [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,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
                     => ~ ( v2_euclid_4(D,A)
                          & v2_euclid_4(E,A)
                          & r2_hidden(B,D)
                          & r2_hidden(C,D)
                          & r2_hidden(B,E)
                          & r2_hidden(C,E)
                          & B != C
                          & D != E ) ) ) ) ) ) )).

fof(t51_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
         => ~ ( v2_euclid_4(B,A)
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
                     => ~ ( r2_hidden(C,B)
                          & r2_hidden(D,B)
                          & C != D ) ) ) ) ) ) )).

fof(t52_euclid_4,axiom,(
    ! [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,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
             => ~ ( v2_euclid_4(C,A)
                  & ! [D] :
                      ( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
                     => ~ ( B != D
                          & r2_hidden(D,C) ) ) ) ) ) ) )).

fof(d9_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => k8_euclid_4(A,B) = B ) ) )).

fof(d10_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => k9_euclid_4(A,B) = k4_euclid_4(A,k8_euclid_4(A,B)) ) ) )).

fof(d11_euclid_4,axiom,(
    ! [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)))
             => k10_euclid_4(A,B,C) = k5_euclid_4(A,k8_euclid_4(A,B),k8_euclid_4(A,C)) ) ) ) )).

fof(d12_euclid_4,axiom,(
    ! [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)))
             => ( r2_euclid_4(A,B,C)
              <=> k10_euclid_4(A,B,C) = np__0 ) ) ) ) )).

fof(symmetry_r1_euclid_4,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k1_euclid(A))
        & m1_subset_1(C,k1_euclid(A)) )
     => ( r1_euclid_4(A,B,C)
       => r1_euclid_4(A,C,B) ) ) )).

fof(symmetry_r2_euclid_4,axiom,(
    ! [A,B,C] :
      ( ( 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))) )
     => ( r2_euclid_4(A,B,C)
       => r2_euclid_4(A,C,B) ) ) )).

fof(dt_k1_euclid_4,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k1_euclid(A))
        & m1_subset_1(C,k1_euclid(A)) )
     => m1_subset_1(k1_euclid_4(A,B,C),k1_zfmisc_1(k1_euclid(A))) ) )).

fof(dt_k2_euclid_4,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k1_euclid(A))
        & m1_subset_1(C,k1_euclid(A)) )
     => m1_subset_1(k2_euclid_4(A,B,C),k1_zfmisc_1(k1_euclid(A))) ) )).

fof(commutativity_k2_euclid_4,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k1_euclid(A))
        & m1_subset_1(C,k1_euclid(A)) )
     => k2_euclid_4(A,B,C) = k2_euclid_4(A,C,B) ) )).

fof(redefinition_k2_euclid_4,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k1_euclid(A))
        & m1_subset_1(C,k1_euclid(A)) )
     => k2_euclid_4(A,B,C) = k1_euclid_4(A,B,C) ) )).

fof(dt_k3_euclid_4,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k1_euclid(A)) )
     => m2_finseq_1(k3_euclid_4(A,B),k1_numbers) ) )).

fof(dt_k4_euclid_4,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k1_euclid(A)) )
     => m1_subset_1(k4_euclid_4(A,B),k1_numbers) ) )).

fof(dt_k5_euclid_4,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k1_euclid(A))
        & m1_subset_1(C,k1_euclid(A)) )
     => m1_subset_1(k5_euclid_4(A,B,C),k1_numbers) ) )).

fof(commutativity_k5_euclid_4,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k1_euclid(A))
        & m1_subset_1(C,k1_euclid(A)) )
     => k5_euclid_4(A,B,C) = k5_euclid_4(A,C,B) ) )).

fof(dt_k6_euclid_4,axiom,(
    ! [A,B,C] :
      ( ( 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(k6_euclid_4(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) )).

fof(dt_k7_euclid_4,axiom,(
    ! [A,B,C] :
      ( ( 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(k7_euclid_4(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) )).

fof(commutativity_k7_euclid_4,axiom,(
    ! [A,B,C] :
      ( ( 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))) )
     => k7_euclid_4(A,B,C) = k7_euclid_4(A,C,B) ) )).

fof(redefinition_k7_euclid_4,axiom,(
    ! [A,B,C] :
      ( ( 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))) )
     => k7_euclid_4(A,B,C) = k6_euclid_4(A,B,C) ) )).

fof(dt_k8_euclid_4,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,u1_struct_0(k15_euclid(A))) )
     => m1_subset_1(k8_euclid_4(A,B),k1_euclid(A)) ) )).

fof(dt_k9_euclid_4,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,u1_struct_0(k15_euclid(A))) )
     => m1_subset_1(k9_euclid_4(A,B),k1_numbers) ) )).

fof(dt_k10_euclid_4,axiom,(
    ! [A,B,C] :
      ( ( 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(k10_euclid_4(A,B,C),k1_numbers) ) )).

fof(commutativity_k10_euclid_4,axiom,(
    ! [A,B,C] :
      ( ( 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))) )
     => k10_euclid_4(A,B,C) = k10_euclid_4(A,C,B) ) )).

fof(d1_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_euclid(A))
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(A))
             => k1_euclid_4(A,B,C) = a_3_0_euclid_4(A,B,C) ) ) ) )).

fof(t45_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
         => ! [C] :
              ( m1_subset_1(C,k1_euclid(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_euclid(A))
                 => ! [E] :
                      ( m1_subset_1(E,k1_euclid(A))
                     => ~ ( B = a_4_0_euclid_4(A,C,D,E)
                          & ! [F] :
                              ( m1_subset_1(F,k1_euclid(A))
                             => ~ ( r2_hidden(F,k2_euclid_4(A,C,D))
                                  & k4_euclid_4(A,k8_euclid(A,E,F)) = k5_seq_4(B)
                                  & r1_euclid_4(A,k8_euclid(A,C,D),k8_euclid(A,E,F)) ) ) ) ) ) ) ) ) )).

fof(d7_euclid_4,axiom,(
    ! [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)))
             => k6_euclid_4(A,B,C) = a_3_1_euclid_4(A,B,C) ) ) ) )).

fof(t53_euclid_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
         => ! [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)))
                     => ~ ( B = a_4_1_euclid_4(A,C,D,E)
                          & ! [F] :
                              ( m1_subset_1(F,u1_struct_0(k15_euclid(A)))
                             => ~ ( r2_hidden(F,k7_euclid_4(A,C,D))
                                  & k9_euclid_4(A,k20_euclid(A,E,F)) = k5_seq_4(B)
                                  & r2_euclid_4(A,k20_euclid(A,C,D),k20_euclid(A,E,F)) ) ) ) ) ) ) ) ) )).

fof(fraenkel_a_3_0_euclid_4,axiom,(
    ! [A,B,C,D] :
      ( ( m2_subset_1(B,k1_numbers,k5_numbers)
        & m1_subset_1(C,k1_euclid(B))
        & m1_subset_1(D,k1_euclid(B)) )
     => ( r2_hidden(A,a_3_0_euclid_4(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,k1_numbers)
            & A = k7_euclid(B,k9_euclid(B,k5_real_1(np__1,E),C),k9_euclid(B,E,D)) ) ) ) )).

fof(fraenkel_a_4_0_euclid_4,axiom,(
    ! [A,B,C,D,E] :
      ( ( m2_subset_1(B,k1_numbers,k5_numbers)
        & m1_subset_1(C,k1_euclid(B))
        & m1_subset_1(D,k1_euclid(B))
        & m1_subset_1(E,k1_euclid(B)) )
     => ( r2_hidden(A,a_4_0_euclid_4(B,C,D,E))
      <=> ? [F] :
            ( m1_subset_1(F,k1_euclid(B))
            & A = k4_euclid_4(B,k8_euclid(B,E,F))
            & r2_hidden(F,k2_euclid_4(B,C,D)) ) ) ) )).

fof(fraenkel_a_3_1_euclid_4,axiom,(
    ! [A,B,C,D] :
      ( ( m2_subset_1(B,k1_numbers,k5_numbers)
        & m1_subset_1(C,u1_struct_0(k15_euclid(B)))
        & m1_subset_1(D,u1_struct_0(k15_euclid(B))) )
     => ( r2_hidden(A,a_3_1_euclid_4(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,k1_numbers)
            & A = k17_euclid(B,k18_euclid(k5_real_1(np__1,E),B,C),k18_euclid(E,B,D)) ) ) ) )).

fof(fraenkel_a_4_1_euclid_4,axiom,(
    ! [A,B,C,D,E] :
      ( ( m2_subset_1(B,k1_numbers,k5_numbers)
        & m1_subset_1(C,u1_struct_0(k15_euclid(B)))
        & m1_subset_1(D,u1_struct_0(k15_euclid(B)))
        & m1_subset_1(E,u1_struct_0(k15_euclid(B))) )
     => ( r2_hidden(A,a_4_1_euclid_4(B,C,D,E))
      <=> ? [F] :
            ( m1_subset_1(F,u1_struct_0(k15_euclid(B)))
            & A = k9_euclid_4(B,k20_euclid(B,E,F))
            & r2_hidden(F,k7_euclid_4(B,C,D)) ) ) ) )).
%------------------------------------------------------------------------------