SET007 Axioms: SET007+402.ax


%------------------------------------------------------------------------------
% File     : SET007+402 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Sequences in 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    : toprns_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   69 (   5 unit)
%            Number of atoms       :  479 (  49 equality)
%            Maximal formula depth :   18 (   9 average)
%            Number of connectives :  429 (  19 ~  ;   1  |; 196  &)
%                                         (  14 <=>; 199 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   16 (   1 propositional; 0-3 arity)
%            Number of functors    :   36 (   6 constant; 0-4 arity)
%            Number of variables   :  203 (   0 singleton; 195 !;   8 ?)
%            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_toprns_1,axiom,(
    $true )).

fof(t2_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( ( v1_funct_1(B)
              & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
              & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
          <=> ( k1_relat_1(B) = k5_numbers
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => m1_subset_1(k1_funct_1(B,C),u1_struct_0(k15_euclid(A))) ) ) ) ) ) )).

fof(d1_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ( v1_toprns_1(B,A)
          <=> r1_tarski(k2_relat_1(B),k4_xboole_0(u1_struct_0(k15_euclid(A)),k1_struct_0(k15_euclid(A),k16_euclid(A)))) ) ) ) )).

fof(t3_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ( v1_toprns_1(B,A)
          <=> ! [C] :
                ~ ( r2_hidden(C,k5_numbers)
                  & k1_funct_1(B,C) = k16_euclid(A) ) ) ) ) )).

fof(t4_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ( v1_toprns_1(B,A)
          <=> ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k2_normsp_1(k15_euclid(A),B,C) != k16_euclid(A) ) ) ) ) )).

fof(t5_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ( ! [D] :
                    ( r2_hidden(D,k5_numbers)
                   => k1_funct_1(B,D) = k1_funct_1(C,D) )
               => B = C ) ) ) ) )).

fof(t6_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ( ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_normsp_1(k15_euclid(A),B,D) = k2_normsp_1(k15_euclid(A),C,D) )
               => B = C ) ) ) ) )).

fof(d2_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(k15_euclid(A)))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(k15_euclid(A))) )
                 => ( D = k1_toprns_1(A,B,C)
                  <=> ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => k2_normsp_1(k15_euclid(A),D,E) = k17_euclid(A,k2_normsp_1(k15_euclid(A),B,E),k2_normsp_1(k15_euclid(A),C,E)) ) ) ) ) ) ) )).

fof(d3_toprns_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(B)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(B))) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(k15_euclid(B)))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(k15_euclid(B))) )
                 => ( D = k2_toprns_1(A,B,C)
                  <=> ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => k2_normsp_1(k15_euclid(B),D,E) = k18_euclid(A,B,k2_normsp_1(k15_euclid(B),C,E)) ) ) ) ) ) ) )).

fof(d4_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ( C = k3_toprns_1(A,B)
              <=> ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_normsp_1(k15_euclid(A),C,D) = k19_euclid(A,k2_normsp_1(k15_euclid(A),B,D)) ) ) ) ) ) )).

fof(d5_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => k4_toprns_1(A,B,C) = k1_toprns_1(A,B,k3_toprns_1(A,C)) ) ) ) )).

fof(d6_toprns_1,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_numbers)
             => ( C = k5_toprns_1(A,B)
              <=> ? [D] :
                    ( m2_finseq_1(D,k1_numbers)
                    & B = D
                    & C = k12_euclid(D) ) ) ) ) ) )).

fof(d7_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( C = k6_toprns_1(A,B)
              <=> ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_seq_1(k5_numbers,k1_numbers,C,D) = k5_toprns_1(A,k2_normsp_1(k15_euclid(A),B,D)) ) ) ) ) ) )).

fof(t7_toprns_1,axiom,(
    $true )).

fof(t8_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => k4_real_1(k18_complex1(B),k5_toprns_1(A,C)) = k5_toprns_1(A,k18_euclid(B,A,C)) ) ) ) )).

fof(t9_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => k6_toprns_1(A,k2_toprns_1(B,A,C)) = k14_seq_1(k6_toprns_1(A,C),k18_complex1(B)) ) ) ) )).

fof(t10_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => k1_toprns_1(A,B,C) = k1_toprns_1(A,C,B) ) ) ) )).

fof(t11_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(k15_euclid(A)))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(k15_euclid(A))) )
                 => k1_toprns_1(A,k1_toprns_1(A,B,C),D) = k1_toprns_1(A,B,k1_toprns_1(A,C,D)) ) ) ) ) )).

fof(t12_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => k3_toprns_1(A,B) = k2_toprns_1(k1_real_1(np__1),A,B) ) ) )).

fof(t13_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(k15_euclid(A)))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(k15_euclid(A))) )
                 => k2_toprns_1(B,A,k1_toprns_1(A,C,D)) = k1_toprns_1(A,k2_toprns_1(B,A,C),k2_toprns_1(B,A,D)) ) ) ) ) )).

fof(t14_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(k15_euclid(A)))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(k15_euclid(A))) )
                 => k2_toprns_1(k4_real_1(B,C),A,D) = k2_toprns_1(B,A,k2_toprns_1(C,A,D)) ) ) ) ) )).

fof(t15_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(k15_euclid(A)))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(k15_euclid(A))) )
                 => k2_toprns_1(B,A,k4_toprns_1(A,C,D)) = k4_toprns_1(A,k2_toprns_1(B,A,C),k2_toprns_1(B,A,D)) ) ) ) ) )).

fof(t16_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(k15_euclid(A)))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(k15_euclid(A))) )
                 => k4_toprns_1(A,B,k1_toprns_1(A,C,D)) = k4_toprns_1(A,k4_toprns_1(A,B,C),D) ) ) ) ) )).

fof(t17_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => k2_toprns_1(np__1,A,B) = B ) ) )).

fof(t18_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => k3_toprns_1(A,k3_toprns_1(A,B)) = B ) ) )).

fof(t19_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => k4_toprns_1(A,B,k3_toprns_1(A,C)) = k1_toprns_1(A,B,C) ) ) ) )).

fof(t20_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(k15_euclid(A)))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(k15_euclid(A))) )
                 => k4_toprns_1(A,B,k4_toprns_1(A,C,D)) = k1_toprns_1(A,k4_toprns_1(A,B,C),D) ) ) ) ) )).

fof(t21_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(k15_euclid(A)))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(k15_euclid(A))) )
                 => k1_toprns_1(A,B,k4_toprns_1(A,C,D)) = k4_toprns_1(A,k1_toprns_1(A,B,C),D) ) ) ) ) )).

fof(t22_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ( v1_toprns_1(C,A)
               => ( B = np__0
                  | v1_toprns_1(k2_toprns_1(B,A,C),A) ) ) ) ) ) )).

fof(t23_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ( v1_toprns_1(B,A)
           => v1_toprns_1(k3_toprns_1(A,B),A) ) ) ) )).

fof(t24_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k5_toprns_1(A,k16_euclid(A)) = np__0 ) )).

fof(t25_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ( k5_toprns_1(A,B) = np__0
           => B = k16_euclid(A) ) ) ) )).

fof(t26_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => r1_xreal_0(np__0,k5_toprns_1(A,B)) ) ) )).

fof(t27_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => k5_toprns_1(A,k19_euclid(A,B)) = k5_toprns_1(A,B) ) ) )).

fof(t28_toprns_1,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)))
             => k5_toprns_1(A,k20_euclid(A,B,C)) = k5_toprns_1(A,k20_euclid(A,C,B)) ) ) ) )).

fof(t29_toprns_1,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)))
             => ( k5_toprns_1(A,k20_euclid(A,B,C)) = np__0
              <=> B = C ) ) ) ) )).

fof(t30_toprns_1,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)))
             => r1_xreal_0(k5_toprns_1(A,k17_euclid(A,B,C)),k3_real_1(k5_toprns_1(A,B),k5_toprns_1(A,C))) ) ) ) )).

fof(t31_toprns_1,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)))
             => r1_xreal_0(k5_toprns_1(A,k20_euclid(A,B,C)),k3_real_1(k5_toprns_1(A,B),k5_toprns_1(A,C))) ) ) ) )).

fof(t32_toprns_1,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)))
             => r1_xreal_0(k5_real_1(k5_toprns_1(A,B),k5_toprns_1(A,C)),k5_toprns_1(A,k17_euclid(A,B,C))) ) ) ) )).

fof(t33_toprns_1,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)))
             => r1_xreal_0(k5_real_1(k5_toprns_1(A,B),k5_toprns_1(A,C)),k5_toprns_1(A,k20_euclid(A,B,C))) ) ) ) )).

fof(t34_toprns_1,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)))
             => ~ ( B != C
                  & r1_xreal_0(k5_toprns_1(A,k20_euclid(A,B,C)),np__0) ) ) ) ) )).

fof(t35_toprns_1,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)))
                 => r1_xreal_0(k5_toprns_1(A,k20_euclid(A,B,C)),k3_real_1(k5_toprns_1(A,k20_euclid(A,B,D)),k5_toprns_1(A,k20_euclid(A,D,C)))) ) ) ) ) )).

fof(t36_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
                     => ~ ( r1_xreal_0(np__0,B)
                          & ~ r1_xreal_0(k5_toprns_1(A,E),k5_toprns_1(A,D))
                          & ~ r1_xreal_0(C,B)
                          & r1_xreal_0(k4_real_1(k5_toprns_1(A,E),C),k4_real_1(k5_toprns_1(A,D),B)) ) ) ) ) ) ) )).

fof(t37_toprns_1,axiom,(
    $true )).

fof(t38_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ( ( ~ r1_xreal_0(B,k1_real_1(k5_toprns_1(A,C)))
                  & ~ r1_xreal_0(k5_toprns_1(A,C),B) )
              <=> ~ r1_xreal_0(k5_toprns_1(A,C),k18_complex1(B)) ) ) ) ) )).

fof(d8_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ( v2_toprns_1(B,A)
          <=> ? [C] :
                ( m1_subset_1(C,k1_numbers)
                & ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ~ r1_xreal_0(C,k5_toprns_1(A,k2_normsp_1(k15_euclid(A),B,D))) ) ) ) ) ) )).

fof(t39_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ? [D] :
                  ( m1_subset_1(D,k1_numbers)
                  & ~ r1_xreal_0(D,np__0)
                  & ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ~ ( r1_xreal_0(E,C)
                          & r1_xreal_0(D,k5_toprns_1(A,k2_normsp_1(k15_euclid(A),B,E))) ) ) ) ) ) ) )).

fof(d9_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ( v3_toprns_1(B,A)
          <=> ? [C] :
                ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
                & ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ~ ( ~ r1_xreal_0(D,np__0)
                        & ! [E] :
                            ( m2_subset_1(E,k1_numbers,k5_numbers)
                           => ? [F] :
                                ( m2_subset_1(F,k1_numbers,k5_numbers)
                                & r1_xreal_0(E,F)
                                & r1_xreal_0(D,k5_toprns_1(A,k20_euclid(A,k2_normsp_1(k15_euclid(A),B,F),C))) ) ) ) ) ) ) ) ) )).

fof(d10_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ( v3_toprns_1(B,A)
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
               => ( C = k7_toprns_1(A,B)
                <=> ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & ! [E] :
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                             => ? [F] :
                                  ( m2_subset_1(F,k1_numbers,k5_numbers)
                                  & r1_xreal_0(E,F)
                                  & r1_xreal_0(D,k5_toprns_1(A,k20_euclid(A,k2_normsp_1(k15_euclid(A),B,F),C))) ) ) ) ) ) ) ) ) ) )).

fof(t40_toprns_1,axiom,(
    $true )).

fof(t41_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ( ( v3_toprns_1(B,A)
                  & v3_toprns_1(C,A) )
               => v3_toprns_1(k1_toprns_1(A,B,C),A) ) ) ) ) )).

fof(t42_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ( ( v3_toprns_1(B,A)
                  & v3_toprns_1(C,A) )
               => k7_toprns_1(A,k1_toprns_1(A,B,C)) = k17_euclid(A,k7_toprns_1(A,B),k7_toprns_1(A,C)) ) ) ) ) )).

fof(t43_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ( v3_toprns_1(C,A)
               => v3_toprns_1(k2_toprns_1(B,A,C),A) ) ) ) ) )).

fof(t44_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ( v3_toprns_1(C,A)
               => k7_toprns_1(A,k2_toprns_1(B,A,C)) = k18_euclid(B,A,k7_toprns_1(A,C)) ) ) ) ) )).

fof(t45_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ( v3_toprns_1(B,A)
           => v3_toprns_1(k3_toprns_1(A,B),A) ) ) ) )).

fof(t46_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ( v3_toprns_1(B,A)
           => k7_toprns_1(A,k3_toprns_1(A,B)) = k19_euclid(A,k7_toprns_1(A,B)) ) ) ) )).

fof(t47_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ( ( v3_toprns_1(B,A)
                  & v3_toprns_1(C,A) )
               => v3_toprns_1(k4_toprns_1(A,B,C),A) ) ) ) ) )).

fof(t48_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => ( ( v3_toprns_1(B,A)
                  & v3_toprns_1(C,A) )
               => k7_toprns_1(A,k4_toprns_1(A,B,C)) = k20_euclid(A,k7_toprns_1(A,B),k7_toprns_1(A,C)) ) ) ) ) )).

fof(t49_toprns_1,axiom,(
    $true )).

fof(t50_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ( v3_toprns_1(B,A)
           => v2_toprns_1(B,A) ) ) ) )).

fof(t51_toprns_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ~ ( v3_toprns_1(B,A)
              & k7_toprns_1(A,B) != k16_euclid(A)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ? [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                      & r1_xreal_0(C,D)
                      & r1_xreal_0(k5_toprns_1(A,k2_normsp_1(k15_euclid(A),B,D)),k6_real_1(k5_toprns_1(A,k7_toprns_1(A,B)),np__2)) ) ) ) ) ) )).

fof(s1_toprns_1,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & v1_funct_2(A,k5_numbers,u1_struct_0(k15_euclid(f1_s1_toprns_1)))
      & m2_relset_1(A,k5_numbers,u1_struct_0(k15_euclid(f1_s1_toprns_1)))
      & ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k2_normsp_1(k15_euclid(f1_s1_toprns_1),A,B) = f2_s1_toprns_1(B) ) ) )).

fof(dt_k1_toprns_1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
        & m1_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A)))
        & v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
        & m1_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
     => ( v1_funct_1(k1_toprns_1(A,B,C))
        & v1_funct_2(k1_toprns_1(A,B,C),k5_numbers,u1_struct_0(k15_euclid(A)))
        & m2_relset_1(k1_toprns_1(A,B,C),k5_numbers,u1_struct_0(k15_euclid(A))) ) ) )).

fof(dt_k2_toprns_1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k5_numbers)
        & v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(B)))
        & m1_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(B))) )
     => ( v1_funct_1(k2_toprns_1(A,B,C))
        & v1_funct_2(k2_toprns_1(A,B,C),k5_numbers,u1_struct_0(k15_euclid(B)))
        & m2_relset_1(k2_toprns_1(A,B,C),k5_numbers,u1_struct_0(k15_euclid(B))) ) ) )).

fof(dt_k3_toprns_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
        & m1_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
     => ( v1_funct_1(k3_toprns_1(A,B))
        & v1_funct_2(k3_toprns_1(A,B),k5_numbers,u1_struct_0(k15_euclid(A)))
        & m2_relset_1(k3_toprns_1(A,B),k5_numbers,u1_struct_0(k15_euclid(A))) ) ) )).

fof(dt_k4_toprns_1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
        & m1_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A)))
        & v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
        & m1_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
     => ( v1_funct_1(k4_toprns_1(A,B,C))
        & v1_funct_2(k4_toprns_1(A,B,C),k5_numbers,u1_struct_0(k15_euclid(A)))
        & m2_relset_1(k4_toprns_1(A,B,C),k5_numbers,u1_struct_0(k15_euclid(A))) ) ) )).

fof(dt_k5_toprns_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,u1_struct_0(k15_euclid(A))) )
     => m1_subset_1(k5_toprns_1(A,B),k1_numbers) ) )).

fof(dt_k6_toprns_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
        & m1_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
     => ( v1_funct_1(k6_toprns_1(A,B))
        & v1_funct_2(k6_toprns_1(A,B),k5_numbers,k1_numbers)
        & m2_relset_1(k6_toprns_1(A,B),k5_numbers,k1_numbers) ) ) )).

fof(dt_k7_toprns_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
        & m1_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
     => m1_subset_1(k7_toprns_1(A,B),u1_struct_0(k15_euclid(A))) ) )).
%------------------------------------------------------------------------------