SET007 Axioms: SET007+133.ax


%------------------------------------------------------------------------------
% File     : SET007+133 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Integer and Rational Exponents
% 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    : prepower [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  138 (  17 unit)
%            Number of atoms       :  664 (  97 equality)
%            Maximal formula depth :   17 (   7 average)
%            Number of connectives :  612 (  86 ~  ;  18  |; 183  &)
%                                         (   7 <=>; 318 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   19 (   1 propositional; 0-3 arity)
%            Number of functors    :   38 (   6 constant; 0-4 arity)
%            Number of variables   :  276 (   0 singleton; 272 !;   4 ?)
%            Maximal term depth    :    4 (   1 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(fc1_prepower,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ( v4_ordinal2(k16_complex1(A))
        & v1_xcmplx_0(k16_complex1(A))
        & v1_xreal_0(k16_complex1(A))
        & ~ v3_xreal_0(k16_complex1(A))
        & v1_int_1(k16_complex1(A))
        & v1_rat_1(k16_complex1(A)) ) ) )).

fof(fc2_prepower,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_int_1(B) )
     => ( v1_xcmplx_0(k6_prepower(A,B))
        & v1_xreal_0(k6_prepower(A,B)) ) ) )).

fof(fc3_prepower,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_rat_1(B) )
     => ( v1_xcmplx_0(k8_prepower(A,B))
        & v1_xreal_0(k8_prepower(A,B)) ) ) )).

fof(rc1_prepower,axiom,(
    ? [A] :
      ( m1_relset_1(A,k5_numbers,k1_numbers)
      & v1_relat_1(A)
      & v1_funct_1(A)
      & v1_funct_2(A,k5_numbers,k1_numbers)
      & v1_seq_1(A)
      & v1_prepower(A) ) )).

fof(t1_prepower,axiom,(
    $true )).

fof(t2_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( ( v4_seq_2(B)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => r1_xreal_0(A,k2_seq_1(k5_numbers,k1_numbers,B,C)) ) )
           => r1_xreal_0(A,k2_seq_2(B)) ) ) ) )).

fof(t3_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( ( v4_seq_2(B)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,B,C),A) ) )
           => r1_xreal_0(k2_seq_2(B),A) ) ) ) )).

fof(d1_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( B = k2_prepower(A)
          <=> ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k2_seq_1(k5_numbers,k1_numbers,B,C) = k2_newton(A,C) ) ) ) ) )).

fof(t4_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( B = k2_prepower(A)
          <=> ( k2_seq_1(k5_numbers,k1_numbers,B,np__0) = np__1
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => k2_seq_1(k5_numbers,k1_numbers,B,k1_nat_1(C,np__1)) = k3_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,B,C),A) ) ) ) ) ) )).

fof(t5_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( A != np__0
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,k2_prepower(A),B) != np__0 ) ) ) )).

fof(t6_prepower,axiom,(
    $true )).

fof(t7_prepower,axiom,(
    $true )).

fof(t8_prepower,axiom,(
    $true )).

fof(t9_prepower,axiom,(
    $true )).

fof(t10_prepower,axiom,(
    $true )).

fof(t11_prepower,axiom,(
    $true )).

fof(t12_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ~ ( np__0 != A
              & np__0 = k2_newton(A,B) ) ) ) )).

fof(t13_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ~ ( ~ r1_xreal_0(A,np__0)
              & r1_xreal_0(k2_newton(A,B),np__0) ) ) ) )).

fof(t14_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => k2_newton(k7_xcmplx_0(np__1,A),B) = k7_xcmplx_0(np__1,k2_newton(A,B)) ) ) )).

fof(t15_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => k2_newton(k7_xcmplx_0(A,B),C) = k7_xcmplx_0(k2_newton(A,C),k2_newton(B,C)) ) ) ) )).

fof(t16_prepower,axiom,(
    $true )).

fof(t17_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => ( r1_xreal_0(A,B)
               => ( r1_xreal_0(A,np__0)
                  | r1_xreal_0(k2_newton(A,C),k2_newton(B,C)) ) ) ) ) ) )).

fof(t18_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => ~ ( r1_xreal_0(np__0,A)
                  & ~ r1_xreal_0(B,A)
                  & r1_xreal_0(np__1,C)
                  & r1_xreal_0(k2_newton(B,C),k2_newton(A,C)) ) ) ) ) )).

fof(t19_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( r1_xreal_0(np__1,A)
           => r1_xreal_0(np__1,k2_newton(A,B)) ) ) ) )).

fof(t20_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( ( r1_xreal_0(np__1,A)
              & r1_xreal_0(np__1,B) )
           => r1_xreal_0(A,k2_newton(A,B)) ) ) ) )).

fof(t21_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ~ ( ~ r1_xreal_0(A,np__1)
              & r1_xreal_0(np__2,B)
              & r1_xreal_0(k2_newton(A,B),A) ) ) ) )).

fof(t22_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( ( r1_xreal_0(A,np__1)
              & r1_xreal_0(np__1,B) )
           => ( r1_xreal_0(A,np__0)
              | r1_xreal_0(k2_newton(A,B),A) ) ) ) ) )).

fof(t23_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ~ ( ~ r1_xreal_0(A,np__0)
              & ~ r1_xreal_0(np__1,A)
              & r1_xreal_0(np__2,B)
              & r1_xreal_0(A,k2_newton(A,B)) ) ) ) )).

fof(t24_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( ~ r1_xreal_0(A,k1_real_1(np__1))
           => r1_xreal_0(k2_xcmplx_0(np__1,k3_xcmplx_0(B,A)),k2_newton(k2_xcmplx_0(np__1,A),B)) ) ) ) )).

fof(t25_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ~ ( ~ r1_xreal_0(A,np__0)
              & ~ r1_xreal_0(np__1,A)
              & ~ r1_xreal_0(k2_newton(k2_xcmplx_0(np__1,A),B),k2_xcmplx_0(np__1,k3_xcmplx_0(k3_newton(np__3,B),A))) ) ) ) )).

fof(t26_prepower,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( ( v4_seq_2(B)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => k2_seq_1(k5_numbers,k1_numbers,C,D) = k3_prepower(k2_seq_1(k5_numbers,k1_numbers,B,D),A) ) )
               => ( v4_seq_2(C)
                  & k2_seq_2(C) = k3_prepower(k2_seq_2(B),A) ) ) ) ) ) )).

fof(d2_prepower,axiom,(
    $true )).

fof(d3_prepower,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( r1_xreal_0(np__1,A)
           => ! [C] :
                ( v1_xreal_0(C)
               => ( ( ~ r1_xreal_0(B,np__0)
                   => ( C = k4_prepower(A,B)
                    <=> ( k2_newton(C,A) = B
                        & ~ r1_xreal_0(C,np__0) ) ) )
                  & ( B = np__0
                   => ( C = k4_prepower(A,B)
                    <=> C = np__0 ) ) ) ) ) ) ) )).

fof(t27_prepower,axiom,(
    $true )).

fof(t28_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ( r1_xreal_0(np__0,A)
              & r1_xreal_0(np__1,B) )
           => ( k2_newton(k4_prepower(B,A),B) = A
              & k4_prepower(B,k2_newton(A,B)) = A ) ) ) ) )).

fof(t29_prepower,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( r1_xreal_0(np__1,A)
       => k5_prepower(A,np__1) = np__1 ) ) )).

fof(t30_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( r1_xreal_0(np__0,A)
       => k4_prepower(np__1,A) = A ) ) )).

fof(t31_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( r1_xreal_0(np__0,A)
                  & r1_xreal_0(np__0,B)
                  & r1_xreal_0(np__1,C) )
               => k4_prepower(C,k3_xcmplx_0(A,B)) = k3_xcmplx_0(k4_prepower(C,A),k4_prepower(C,B)) ) ) ) ) )).

fof(t32_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(np__1,B)
           => ( r1_xreal_0(A,np__0)
              | k4_prepower(B,k7_xcmplx_0(np__1,A)) = k7_xcmplx_0(np__1,k4_prepower(B,A)) ) ) ) ) )).

fof(t33_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( r1_xreal_0(np__0,A)
                  & r1_xreal_0(np__1,C) )
               => ( r1_xreal_0(B,np__0)
                  | k4_prepower(C,k7_xcmplx_0(A,B)) = k7_xcmplx_0(k4_prepower(C,A),k4_prepower(C,B)) ) ) ) ) ) )).

fof(t34_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( r1_xreal_0(np__0,A)
                  & r1_xreal_0(np__1,B)
                  & r1_xreal_0(np__1,C) )
               => k4_prepower(B,k4_prepower(C,A)) = k4_prepower(k2_nat_1(B,C),A) ) ) ) ) )).

fof(t35_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( r1_xreal_0(np__0,A)
                  & r1_xreal_0(np__1,B)
                  & r1_xreal_0(np__1,C) )
               => k3_xcmplx_0(k4_prepower(B,A),k4_prepower(C,A)) = k4_prepower(k2_nat_1(B,C),k2_newton(A,k1_nat_1(B,C))) ) ) ) ) )).

fof(t36_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( r1_xreal_0(np__0,A)
                  & r1_xreal_0(A,B)
                  & r1_xreal_0(np__1,C) )
               => r1_xreal_0(k4_prepower(C,A),k4_prepower(C,B)) ) ) ) ) )).

fof(t37_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ~ ( r1_xreal_0(np__0,A)
                  & ~ r1_xreal_0(B,A)
                  & r1_xreal_0(np__1,C)
                  & r1_xreal_0(k4_prepower(C,B),k4_prepower(C,A)) ) ) ) ) )).

fof(t38_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ( r1_xreal_0(np__1,A)
              & r1_xreal_0(np__1,B) )
           => ( r1_xreal_0(np__1,k4_prepower(B,A))
              & r1_xreal_0(k4_prepower(B,A),A) ) ) ) ) )).

fof(t39_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ( r1_xreal_0(np__0,A)
              & r1_xreal_0(np__1,B) )
           => ( r1_xreal_0(np__1,A)
              | ( r1_xreal_0(A,k4_prepower(B,A))
                & ~ r1_xreal_0(np__1,k4_prepower(B,A)) ) ) ) ) ) )).

fof(t40_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(np__1,B)
           => ( r1_xreal_0(A,np__0)
              | r1_xreal_0(k6_xcmplx_0(k4_prepower(B,A),np__1),k7_xcmplx_0(k6_xcmplx_0(A,np__1),B)) ) ) ) ) )).

fof(t41_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( r1_xreal_0(np__0,A)
       => k4_prepower(np__2,A) = k8_square_1(A) ) ) )).

fof(t42_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( r1_xreal_0(np__1,C)
                 => k2_seq_1(k5_numbers,k1_numbers,B,C) = k4_prepower(C,A) ) )
           => ( r1_xreal_0(A,np__0)
              | ( v4_seq_2(B)
                & k2_seq_2(B) = np__1 ) ) ) ) ) )).

fof(d4_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_int_1(B)
         => ( ( r1_xreal_0(np__0,B)
             => k6_prepower(A,B) = k2_newton(A,k1_prepower(B)) )
            & ( ~ r1_xreal_0(np__0,B)
             => k6_prepower(A,B) = k5_xcmplx_0(k2_newton(A,k1_prepower(B))) ) ) ) ) )).

fof(t43_prepower,axiom,(
    $true )).

fof(t44_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k6_prepower(A,np__0) = np__1 ) )).

fof(t45_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k6_prepower(A,np__1) = A ) )).

fof(t46_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => k6_prepower(A,B) = k2_newton(A,B) ) ) )).

fof(t47_prepower,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => k7_prepower(np__1,A) = np__1 ) )).

fof(t48_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_int_1(B)
         => ~ ( A != np__0
              & k6_prepower(A,B) = np__0 ) ) ) )).

fof(t49_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_int_1(B)
         => ~ ( ~ r1_xreal_0(A,np__0)
              & r1_xreal_0(k6_prepower(A,B),np__0) ) ) ) )).

fof(t50_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_int_1(C)
             => k6_prepower(k3_xcmplx_0(A,B),C) = k3_xcmplx_0(k6_prepower(A,C),k6_prepower(B,C)) ) ) ) )).

fof(t51_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_int_1(B)
         => ( A != np__0
           => k6_prepower(A,k4_xcmplx_0(B)) = k7_xcmplx_0(np__1,k6_prepower(A,B)) ) ) ) )).

fof(t52_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_int_1(B)
         => k6_prepower(k7_xcmplx_0(np__1,A),B) = k7_xcmplx_0(np__1,k6_prepower(A,B)) ) ) )).

fof(t53_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => ( A != np__0
               => k6_prepower(A,k6_xcmplx_0(B,C)) = k7_xcmplx_0(k2_newton(A,B),k2_newton(A,C)) ) ) ) ) )).

fof(t54_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ( A != np__0
               => k6_prepower(A,k2_xcmplx_0(B,C)) = k3_xcmplx_0(k6_prepower(A,B),k6_prepower(A,C)) ) ) ) ) )).

fof(t55_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => k6_prepower(k6_prepower(A,B),C) = k6_prepower(A,k3_xcmplx_0(B,C)) ) ) ) )).

fof(t56_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( v1_int_1(C)
             => ( r1_xreal_0(np__1,B)
               => ( r1_xreal_0(A,np__0)
                  | k6_prepower(k4_prepower(B,A),C) = k4_prepower(B,k6_prepower(A,C)) ) ) ) ) ) )).

fof(d5_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_rat_1(B)
         => k8_prepower(A,B) = k4_prepower(k1_rat_1(B),k6_prepower(A,k2_rat_1(B))) ) ) )).

fof(t57_prepower,axiom,(
    $true )).

fof(t58_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_rat_1(B)
         => ( B = np__0
           => ( r1_xreal_0(A,np__0)
              | k8_prepower(A,B) = np__1 ) ) ) ) )).

fof(t59_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_rat_1(B)
         => ( B = np__1
           => ( r1_xreal_0(A,np__0)
              | k8_prepower(A,B) = A ) ) ) ) )).

fof(t60_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( v1_rat_1(C)
             => ( C = B
               => ( r1_xreal_0(A,np__0)
                  | k8_prepower(A,C) = k2_newton(A,B) ) ) ) ) ) )).

fof(t61_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( v1_rat_1(C)
             => ( ( r1_xreal_0(np__1,B)
                  & C = k2_real_1(B) )
               => ( r1_xreal_0(A,np__0)
                  | k8_prepower(A,C) = k4_prepower(B,A) ) ) ) ) ) )).

fof(t62_prepower,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => k9_prepower(np__1,A) = np__1 ) )).

fof(t63_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_rat_1(B)
         => ~ ( ~ r1_xreal_0(A,np__0)
              & r1_xreal_0(k8_prepower(A,B),np__0) ) ) ) )).

fof(t64_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_rat_1(B)
         => ! [C] :
              ( v1_rat_1(C)
             => ( ~ r1_xreal_0(A,np__0)
               => k3_xcmplx_0(k8_prepower(A,B),k8_prepower(A,C)) = k8_prepower(A,k2_xcmplx_0(B,C)) ) ) ) ) )).

fof(t65_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_rat_1(B)
         => ( ~ r1_xreal_0(A,np__0)
           => k7_xcmplx_0(np__1,k8_prepower(A,B)) = k8_prepower(A,k4_xcmplx_0(B)) ) ) ) )).

fof(t66_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_rat_1(B)
         => ! [C] :
              ( v1_rat_1(C)
             => ( ~ r1_xreal_0(A,np__0)
               => k7_xcmplx_0(k8_prepower(A,B),k8_prepower(A,C)) = k8_prepower(A,k6_xcmplx_0(B,C)) ) ) ) ) )).

fof(t67_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_rat_1(C)
             => ~ ( ~ r1_xreal_0(A,np__0)
                  & ~ r1_xreal_0(B,np__0)
                  & k8_prepower(k3_xcmplx_0(A,B),C) != k3_xcmplx_0(k8_prepower(A,C),k8_prepower(B,C)) ) ) ) ) )).

fof(t68_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_rat_1(B)
         => ( ~ r1_xreal_0(A,np__0)
           => k8_prepower(k7_xcmplx_0(np__1,A),B) = k7_xcmplx_0(np__1,k8_prepower(A,B)) ) ) ) )).

fof(t69_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_rat_1(C)
             => ~ ( ~ r1_xreal_0(A,np__0)
                  & ~ r1_xreal_0(B,np__0)
                  & k8_prepower(k7_xcmplx_0(A,B),C) != k7_xcmplx_0(k8_prepower(A,C),k8_prepower(B,C)) ) ) ) ) )).

fof(t70_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_rat_1(B)
         => ! [C] :
              ( v1_rat_1(C)
             => ( ~ r1_xreal_0(A,np__0)
               => k8_prepower(k8_prepower(A,B),C) = k8_prepower(A,k3_xcmplx_0(B,C)) ) ) ) ) )).

fof(t71_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_rat_1(B)
         => ( ( r1_xreal_0(np__1,A)
              & r1_xreal_0(np__0,B) )
           => r1_xreal_0(np__1,k8_prepower(A,B)) ) ) ) )).

fof(t72_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_rat_1(B)
         => ( ( r1_xreal_0(np__1,A)
              & r1_xreal_0(B,np__0) )
           => r1_xreal_0(k8_prepower(A,B),np__1) ) ) ) )).

fof(t73_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_rat_1(B)
         => ~ ( ~ r1_xreal_0(A,np__1)
              & ~ r1_xreal_0(B,np__0)
              & r1_xreal_0(k8_prepower(A,B),np__1) ) ) ) )).

fof(t74_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_rat_1(B)
         => ! [C] :
              ( v1_rat_1(C)
             => ( ( r1_xreal_0(np__1,A)
                  & r1_xreal_0(C,B) )
               => r1_xreal_0(k8_prepower(A,C),k8_prepower(A,B)) ) ) ) ) )).

fof(t75_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_rat_1(B)
         => ! [C] :
              ( v1_rat_1(C)
             => ~ ( ~ r1_xreal_0(A,np__1)
                  & ~ r1_xreal_0(B,C)
                  & r1_xreal_0(k8_prepower(A,B),k8_prepower(A,C)) ) ) ) ) )).

fof(t76_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_rat_1(B)
         => ~ ( ~ r1_xreal_0(A,np__0)
              & ~ r1_xreal_0(np__1,A)
              & ~ r1_xreal_0(B,np__0)
              & r1_xreal_0(np__1,k8_prepower(A,B)) ) ) ) )).

fof(t77_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_rat_1(B)
         => ( ( r1_xreal_0(A,np__1)
              & r1_xreal_0(B,np__0) )
           => ( r1_xreal_0(A,np__0)
              | r1_xreal_0(np__1,k8_prepower(A,B)) ) ) ) ) )).

fof(d6_prepower,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v1_prepower(A)
      <=> ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => v1_rat_1(k2_seq_1(k5_numbers,k1_numbers,A,B)) ) ) ) )).

fof(t78_prepower,axiom,(
    $true )).

fof(t79_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ? [B] :
          ( v1_funct_1(B)
          & v1_funct_2(B,k5_numbers,k1_numbers)
          & v1_prepower(B)
          & m2_relset_1(B,k5_numbers,k1_numbers)
          & v4_seq_2(B)
          & k2_seq_2(B) = A
          & ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => r1_xreal_0(k10_prepower(B,C),A) ) ) ) )).

fof(t80_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ? [B] :
          ( v1_funct_1(B)
          & v1_funct_2(B,k5_numbers,k1_numbers)
          & v1_prepower(B)
          & m2_relset_1(B,k5_numbers,k1_numbers)
          & v4_seq_2(B)
          & k2_seq_2(B) = A
          & ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => r1_xreal_0(A,k10_prepower(B,C)) ) ) ) )).

fof(d7_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v1_prepower(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( C = k11_prepower(A,B)
              <=> ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_seq_1(k5_numbers,k1_numbers,C,D) = k8_prepower(A,k10_prepower(B,D)) ) ) ) ) ) )).

fof(t81_prepower,axiom,(
    $true )).

fof(t82_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v1_prepower(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( v4_seq_2(B)
           => ( r1_xreal_0(A,np__0)
              | v4_seq_2(k11_prepower(A,B)) ) ) ) ) )).

fof(t83_prepower,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v1_prepower(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v1_prepower(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( v1_xreal_0(C)
             => ( ( v4_seq_2(A)
                  & v4_seq_2(B)
                  & k2_seq_2(A) = k2_seq_2(B) )
               => ( r1_xreal_0(C,np__0)
                  | ( v4_seq_2(k11_prepower(C,A))
                    & v4_seq_2(k11_prepower(C,B))
                    & k2_seq_2(k11_prepower(C,A)) = k2_seq_2(k11_prepower(C,B)) ) ) ) ) ) ) )).

fof(d8_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ~ r1_xreal_0(A,np__0)
           => ! [C] :
                ( v1_xreal_0(C)
               => ( C = k12_prepower(A,B)
                <=> ? [D] :
                      ( v1_funct_1(D)
                      & v1_funct_2(D,k5_numbers,k1_numbers)
                      & v1_prepower(D)
                      & m2_relset_1(D,k5_numbers,k1_numbers)
                      & v4_seq_2(D)
                      & k2_seq_2(D) = B
                      & v4_seq_2(k11_prepower(A,D))
                      & k2_seq_2(k11_prepower(A,D)) = C ) ) ) ) ) ) )).

fof(t84_prepower,axiom,(
    $true )).

fof(t85_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( ~ r1_xreal_0(A,np__0)
       => k12_prepower(A,np__0) = np__1 ) ) )).

fof(t86_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( ~ r1_xreal_0(A,np__0)
       => k12_prepower(A,np__1) = A ) ) )).

fof(t87_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k12_prepower(np__1,A) = np__1 ) )).

fof(t88_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_rat_1(B)
         => ( ~ r1_xreal_0(A,np__0)
           => k12_prepower(A,B) = k8_prepower(A,B) ) ) ) )).

fof(t89_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( ~ r1_xreal_0(A,np__0)
               => k12_prepower(A,k2_xcmplx_0(B,C)) = k3_xcmplx_0(k12_prepower(A,B),k12_prepower(A,C)) ) ) ) ) )).

fof(t90_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ~ r1_xreal_0(A,np__0)
           => k12_prepower(A,k4_xcmplx_0(B)) = k7_xcmplx_0(np__1,k12_prepower(A,B)) ) ) ) )).

fof(t91_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( ~ r1_xreal_0(A,np__0)
               => k12_prepower(A,k6_xcmplx_0(B,C)) = k7_xcmplx_0(k12_prepower(A,B),k12_prepower(A,C)) ) ) ) ) )).

fof(t92_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ~ ( ~ r1_xreal_0(A,np__0)
                  & ~ r1_xreal_0(B,np__0)
                  & k12_prepower(k3_xcmplx_0(A,B),C) != k3_xcmplx_0(k12_prepower(A,C),k12_prepower(B,C)) ) ) ) ) )).

fof(t93_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ~ r1_xreal_0(A,np__0)
           => k12_prepower(k7_xcmplx_0(np__1,A),B) = k7_xcmplx_0(np__1,k12_prepower(A,B)) ) ) ) )).

fof(t94_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ~ ( ~ r1_xreal_0(A,np__0)
                  & ~ r1_xreal_0(B,np__0)
                  & k12_prepower(k7_xcmplx_0(A,B),C) != k7_xcmplx_0(k12_prepower(A,C),k12_prepower(B,C)) ) ) ) ) )).

fof(t95_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ~ ( ~ r1_xreal_0(A,np__0)
              & r1_xreal_0(k12_prepower(A,B),np__0) ) ) ) )).

fof(t96_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( ( r1_xreal_0(np__1,A)
                  & r1_xreal_0(C,B) )
               => r1_xreal_0(k12_prepower(A,C),k12_prepower(A,B)) ) ) ) ) )).

fof(t97_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ~ ( ~ r1_xreal_0(A,np__1)
                  & ~ r1_xreal_0(B,C)
                  & r1_xreal_0(k12_prepower(A,B),k12_prepower(A,C)) ) ) ) ) )).

fof(t98_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( ( r1_xreal_0(A,np__1)
                  & r1_xreal_0(C,B) )
               => ( r1_xreal_0(A,np__0)
                  | r1_xreal_0(k12_prepower(A,B),k12_prepower(A,C)) ) ) ) ) ) )).

fof(t99_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ( r1_xreal_0(np__1,A)
              & r1_xreal_0(np__0,B) )
           => r1_xreal_0(np__1,k12_prepower(A,B)) ) ) ) )).

fof(t100_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ~ ( ~ r1_xreal_0(A,np__1)
              & ~ r1_xreal_0(B,np__0)
              & r1_xreal_0(k12_prepower(A,B),np__1) ) ) ) )).

fof(t101_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ( r1_xreal_0(np__1,A)
              & r1_xreal_0(B,np__0) )
           => r1_xreal_0(k12_prepower(A,B),np__1) ) ) ) )).

fof(t102_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ~ ( ~ r1_xreal_0(A,np__1)
              & ~ r1_xreal_0(np__0,B)
              & r1_xreal_0(np__1,k12_prepower(A,B)) ) ) ) )).

fof(t103_prepower,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( ( v4_seq_2(B)
                  & v4_seq_2(C)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,B,D),np__0)
                        & k2_seq_1(k5_numbers,k1_numbers,C,D) = k9_prepower(k2_seq_1(k5_numbers,k1_numbers,B,D),A) ) ) )
               => ( r1_xreal_0(k2_seq_2(B),np__0)
                  | k2_seq_2(C) = k9_prepower(k2_seq_2(B),A) ) ) ) ) ) )).

fof(t104_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( ( v4_seq_2(B)
                  & v4_seq_2(C)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => k2_seq_1(k5_numbers,k1_numbers,C,D) = k12_prepower(A,k2_seq_1(k5_numbers,k1_numbers,B,D)) ) )
               => ( r1_xreal_0(A,np__0)
                  | k2_seq_2(C) = k12_prepower(A,k2_seq_2(B)) ) ) ) ) ) )).

fof(t105_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( ~ r1_xreal_0(A,np__0)
               => k12_prepower(k12_prepower(A,B),C) = k12_prepower(A,k3_xcmplx_0(B,C)) ) ) ) ) )).

fof(dt_k1_prepower,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => m2_subset_1(k1_prepower(A),k1_numbers,k5_numbers) ) )).

fof(projectivity_k1_prepower,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => k1_prepower(k1_prepower(A)) = k1_prepower(A) ) )).

fof(redefinition_k1_prepower,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => k1_prepower(A) = k16_complex1(A) ) )).

fof(dt_k2_prepower,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( v1_funct_1(k2_prepower(A))
        & v1_funct_2(k2_prepower(A),k5_numbers,k1_numbers)
        & m2_relset_1(k2_prepower(A),k5_numbers,k1_numbers) ) ) )).

fof(dt_k3_prepower,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k5_numbers) )
     => m1_subset_1(k3_prepower(A,B),k1_numbers) ) )).

fof(redefinition_k3_prepower,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k5_numbers) )
     => k3_prepower(A,B) = k2_newton(A,B) ) )).

fof(dt_k4_prepower,axiom,(
    ! [A,B] :
      ( ( v4_ordinal2(A)
        & v1_xreal_0(B) )
     => v1_xreal_0(k4_prepower(A,B)) ) )).

fof(dt_k5_prepower,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k1_numbers) )
     => m1_subset_1(k5_prepower(A,B),k1_numbers) ) )).

fof(redefinition_k5_prepower,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k1_numbers) )
     => k5_prepower(A,B) = k4_prepower(A,B) ) )).

fof(dt_k6_prepower,axiom,(
    $true )).

fof(dt_k7_prepower,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & v1_int_1(B) )
     => m1_subset_1(k7_prepower(A,B),k1_numbers) ) )).

fof(redefinition_k7_prepower,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & v1_int_1(B) )
     => k7_prepower(A,B) = k6_prepower(A,B) ) )).

fof(dt_k8_prepower,axiom,(
    $true )).

fof(dt_k9_prepower,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & v1_rat_1(B) )
     => m1_subset_1(k9_prepower(A,B),k1_numbers) ) )).

fof(redefinition_k9_prepower,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & v1_rat_1(B) )
     => k9_prepower(A,B) = k8_prepower(A,B) ) )).

fof(dt_k10_prepower,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v1_prepower(A)
        & m1_relset_1(A,k5_numbers,k1_numbers)
        & m1_subset_1(B,k5_numbers) )
     => v1_rat_1(k10_prepower(A,B)) ) )).

fof(redefinition_k10_prepower,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v1_prepower(A)
        & m1_relset_1(A,k5_numbers,k1_numbers)
        & m1_subset_1(B,k5_numbers) )
     => k10_prepower(A,B) = k1_funct_1(A,B) ) )).

fof(dt_k11_prepower,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & v1_prepower(B)
        & m1_relset_1(B,k5_numbers,k1_numbers) )
     => ( v1_funct_1(k11_prepower(A,B))
        & v1_funct_2(k11_prepower(A,B),k5_numbers,k1_numbers)
        & m2_relset_1(k11_prepower(A,B),k5_numbers,k1_numbers) ) ) )).

fof(dt_k12_prepower,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => v1_xreal_0(k12_prepower(A,B)) ) )).

fof(dt_k13_prepower,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => m1_subset_1(k13_prepower(A,B),k1_numbers) ) )).

fof(redefinition_k13_prepower,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => k13_prepower(A,B) = k12_prepower(A,B) ) )).
%------------------------------------------------------------------------------