SET007 Axioms: SET007+144.ax


%------------------------------------------------------------------------------
% File     : SET007+144 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Real Exponents and Logarithms
% 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    : power [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   83 (   9 unit)
%            Number of atoms       :  434 ( 104 equality)
%            Maximal formula depth :   16 (   8 average)
%            Number of connectives :  482 ( 131 ~  ;  11  |; 126  &)
%                                         (   6 <=>; 208 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   12 (   1 propositional; 0-3 arity)
%            Number of functors    :   31 (   7 constant; 0-4 arity)
%            Number of variables   :  185 (   0 singleton; 175 !;  10 ?)
%            Maximal term depth    :    5 (   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(t1_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ? [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
                & B = k2_nat_1(np__2,C) )
           => k2_newton(k4_xcmplx_0(A),B) = k2_newton(A,B) ) ) ) )).

fof(t2_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ? [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
                & B = k1_nat_1(k2_nat_1(np__2,C),np__1) )
           => k2_newton(k4_xcmplx_0(A),B) = k4_xcmplx_0(k2_newton(A,B)) ) ) ) )).

fof(t3_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ~ ( ~ r1_xreal_0(np__0,A)
                & ! [C] :
                    ( m2_subset_1(C,k1_numbers,k5_numbers)
                   => B != k2_nat_1(np__2,C) ) )
           => r1_xreal_0(np__0,k2_newton(A,B)) ) ) ) )).

fof(d1_power,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ( ( r1_xreal_0(np__0,B)
                & r1_xreal_0(np__1,A) )
             => k1_power(A,B) = k4_prepower(A,B) )
            & ~ ( ~ r1_xreal_0(np__0,B)
                & ? [C] :
                    ( m2_subset_1(C,k1_numbers,k5_numbers)
                    & A = k1_nat_1(k2_nat_1(np__2,C),np__1) )
                & k1_power(A,B) != k4_xcmplx_0(k4_prepower(A,k4_xcmplx_0(B))) ) ) ) ) )).

fof(t4_power,axiom,(
    $true )).

fof(t5_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ~ ( ~ ( r1_xreal_0(np__1,B)
                    & r1_xreal_0(np__0,A) )
                & ! [C] :
                    ( m2_subset_1(C,k1_numbers,k5_numbers)
                   => B != k1_nat_1(k2_nat_1(np__2,C),np__1) ) )
           => ( k2_newton(k1_power(B,A),B) = A
              & k1_power(B,k2_newton(A,B)) = A ) ) ) ) )).

fof(t6_power,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( r1_xreal_0(np__1,A)
       => k2_power(A,np__0) = np__0 ) ) )).

fof(t7_power,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( r1_xreal_0(np__1,A)
       => k2_power(A,np__1) = np__1 ) ) )).

fof(t8_power,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__0,k1_power(B,A)) ) ) ) )).

fof(t9_power,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( ? [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
            & A = k1_nat_1(k2_nat_1(np__2,B),np__1) )
       => k2_power(A,k1_real_1(np__1)) = k1_real_1(np__1) ) ) )).

fof(t10_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k1_power(np__1,A) = A ) )).

fof(t11_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ? [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
                & B = k1_nat_1(k2_nat_1(np__2,C),np__1) )
           => k1_power(B,A) = k4_xcmplx_0(k1_power(B,k4_xcmplx_0(A))) ) ) ) )).

fof(t12_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ~ ( ~ ( r1_xreal_0(np__1,C)
                        & r1_xreal_0(np__0,A)
                        & r1_xreal_0(np__0,B) )
                    & ! [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                       => C != k1_nat_1(k2_nat_1(np__2,D),np__1) ) )
               => k1_power(C,k3_xcmplx_0(A,B)) = k3_xcmplx_0(k1_power(C,A),k1_power(C,B)) ) ) ) ) )).

fof(t13_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ( ( ~ r1_xreal_0(A,np__0)
                & r1_xreal_0(np__1,B) )
              | ( A != np__0
                & ? [C] :
                    ( m2_subset_1(C,k1_numbers,k5_numbers)
                    & B = k1_nat_1(k2_nat_1(np__2,C),np__1) ) ) )
           => k1_power(B,k7_xcmplx_0(np__1,A)) = k7_xcmplx_0(np__1,k1_power(B,A)) ) ) ) )).

fof(t14_power,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,np__0)
                    & r1_xreal_0(np__1,C) )
                  | ( B != np__0
                    & ? [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                        & C = k1_nat_1(k2_nat_1(np__2,D),np__1) ) ) )
               => k1_power(C,k7_xcmplx_0(A,B)) = k7_xcmplx_0(k1_power(C,A),k1_power(C,B)) ) ) ) ) )).

fof(t15_power,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) )
                    & ! [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                       => ! [E] :
                            ( m2_subset_1(E,k1_numbers,k5_numbers)
                           => ~ ( B = k1_nat_1(k2_nat_1(np__2,D),np__1)
                                & C = k1_nat_1(k2_nat_1(np__2,E),np__1) ) ) ) )
               => k1_power(B,k1_power(C,A)) = k1_power(k2_nat_1(B,C),A) ) ) ) ) )).

fof(t16_power,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) )
                    & ! [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                       => ! [E] :
                            ( m2_subset_1(E,k1_numbers,k5_numbers)
                           => ~ ( B = k1_nat_1(k2_nat_1(np__2,D),np__1)
                                & C = k1_nat_1(k2_nat_1(np__2,E),np__1) ) ) ) )
               => k3_xcmplx_0(k1_power(B,A),k1_power(C,A)) = k1_power(k2_nat_1(B,C),k2_newton(A,k1_nat_1(B,C))) ) ) ) ) )).

fof(t17_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r1_xreal_0(A,B)
               => ( ( ~ ( r1_xreal_0(np__0,A)
                        & r1_xreal_0(np__1,C) )
                    & ! [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                       => C != k1_nat_1(k2_nat_1(np__2,D),np__1) ) )
                  | r1_xreal_0(k1_power(C,A),k1_power(C,B)) ) ) ) ) ) )).

fof(t18_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ~ ( ~ r1_xreal_0(B,A)
                  & ~ ( ~ ( r1_xreal_0(np__0,A)
                          & r1_xreal_0(np__1,C) )
                      & ! [D] :
                          ( m2_subset_1(D,k1_numbers,k5_numbers)
                         => C != k1_nat_1(k2_nat_1(np__2,D),np__1) ) )
                  & r1_xreal_0(k1_power(C,B),k1_power(C,A)) ) ) ) ) )).

fof(t19_power,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,k1_power(B,A))
              & r1_xreal_0(k1_power(B,A),A) ) ) ) ) )).

fof(t20_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(A,k1_real_1(np__1))
           => ( ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => B != k1_nat_1(k2_nat_1(np__2,C),np__1) )
              | ( r1_xreal_0(k1_power(B,A),k1_real_1(np__1))
                & r1_xreal_0(A,k1_power(B,A)) ) ) ) ) ) )).

fof(t21_power,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,k1_power(B,A))
                & ~ r1_xreal_0(np__1,k1_power(B,A)) ) ) ) ) ) )).

fof(t22_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(A,np__0)
           => ( r1_xreal_0(A,k1_real_1(np__1))
              | ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => B != k1_nat_1(k2_nat_1(np__2,C),np__1) )
              | ( r1_xreal_0(k1_power(B,A),A)
                & ~ r1_xreal_0(k1_power(B,A),k1_real_1(np__1)) ) ) ) ) ) )).

fof(t23_power,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(k1_power(B,A),np__1),k7_xcmplx_0(k6_xcmplx_0(A,np__1),B)) ) ) ) ) )).

fof(t24_power,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( r1_xreal_0(np__1,C)
                 => k2_seq_1(k5_numbers,k1_numbers,A,C) = k1_power(C,B) ) )
           => ( r1_xreal_0(B,np__0)
              | ( v4_seq_2(A)
                & k2_seq_2(A) = np__1 ) ) ) ) ) )).

fof(d2_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( ( ~ r1_xreal_0(A,np__0)
                 => ( C = k3_power(A,B)
                  <=> C = k12_prepower(A,B) ) )
                & ( A = np__0
                 => ( r1_xreal_0(B,np__0)
                    | ( C = k3_power(A,B)
                    <=> C = np__0 ) ) )
                & ( ( A = np__0
                    & B = np__0 )
                 => ( C = k3_power(A,B)
                  <=> C = np__1 ) )
                & ( v1_int_1(B)
                 => ( r1_xreal_0(np__0,A)
                    | ( C = k3_power(A,B)
                    <=> ? [D] :
                          ( v1_int_1(D)
                          & D = B
                          & C = k6_prepower(A,D) ) ) ) ) ) ) ) ) )).

fof(t25_power,axiom,(
    $true )).

fof(t26_power,axiom,(
    $true )).

fof(t27_power,axiom,(
    $true )).

fof(t28_power,axiom,(
    $true )).

fof(t29_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k3_power(A,np__0) = np__1 ) )).

fof(t30_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k3_power(A,np__1) = A ) )).

fof(t31_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k3_power(np__1,A) = np__1 ) )).

fof(t32_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( ~ r1_xreal_0(A,np__0)
               => k3_power(A,k2_xcmplx_0(B,C)) = k3_xcmplx_0(k3_power(A,B),k3_power(A,C)) ) ) ) ) )).

fof(t33_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ~ r1_xreal_0(A,np__0)
           => k3_power(A,k4_xcmplx_0(B)) = k7_xcmplx_0(np__1,k3_power(A,B)) ) ) ) )).

fof(t34_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( ~ r1_xreal_0(A,np__0)
               => k3_power(A,k6_xcmplx_0(B,C)) = k7_xcmplx_0(k3_power(A,B),k3_power(A,C)) ) ) ) ) )).

fof(t35_power,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)
                  & k3_power(k3_xcmplx_0(A,B),C) != k3_xcmplx_0(k3_power(A,C),k3_power(B,C)) ) ) ) ) )).

fof(t36_power,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)
                  & k3_power(k7_xcmplx_0(A,B),C) != k7_xcmplx_0(k3_power(A,C),k3_power(B,C)) ) ) ) ) )).

fof(t37_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ~ r1_xreal_0(A,np__0)
           => k3_power(k7_xcmplx_0(np__1,A),B) = k3_power(A,k4_xcmplx_0(B)) ) ) ) )).

fof(t38_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( ~ r1_xreal_0(A,np__0)
               => k3_power(k3_power(A,B),C) = k3_power(A,k3_xcmplx_0(B,C)) ) ) ) ) )).

fof(t39_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ~ ( ~ r1_xreal_0(A,np__0)
              & r1_xreal_0(k3_power(A,B),np__0) ) ) ) )).

fof(t40_power,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(k3_power(A,B),np__1) ) ) ) )).

fof(t41_power,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,k3_power(A,B)) ) ) ) )).

fof(t42_power,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,A)
                  & ~ r1_xreal_0(C,np__0)
                  & r1_xreal_0(k3_power(B,C),k3_power(A,C)) ) ) ) ) )).

fof(t43_power,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,A)
                  & ~ r1_xreal_0(np__0,C)
                  & r1_xreal_0(k3_power(A,C),k3_power(B,C)) ) ) ) ) )).

fof(t44_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ~ ( ~ r1_xreal_0(B,A)
                  & ~ r1_xreal_0(C,np__1)
                  & r1_xreal_0(k3_power(C,B),k3_power(C,A)) ) ) ) ) )).

fof(t45_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ~ ( ~ r1_xreal_0(B,A)
                  & ~ r1_xreal_0(C,np__0)
                  & ~ r1_xreal_0(np__1,C)
                  & r1_xreal_0(k3_power(C,A),k3_power(C,B)) ) ) ) ) )).

fof(t46_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( A != np__0
           => k3_power(A,B) = k2_newton(A,B) ) ) ) )).

fof(t47_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(np__1,B)
           => k3_power(A,B) = k2_newton(A,B) ) ) ) )).

fof(t48_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( A != np__0
           => k3_power(A,B) = k2_newton(A,B) ) ) ) )).

fof(t49_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(np__1,B)
           => k3_power(A,B) = k2_newton(A,B) ) ) ) )).

fof(t50_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_int_1(B)
         => ( A != np__0
           => k3_power(A,B) = k6_prepower(A,B) ) ) ) )).

fof(t51_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_rat_1(B)
         => ( ~ r1_xreal_0(A,np__0)
           => k3_power(A,B) = k8_prepower(A,B) ) ) ) )).

fof(t52_power,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) )
           => k3_power(A,k6_real_1(np__1,B)) = k1_power(B,A) ) ) ) )).

fof(t53_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k3_power(A,np__2) = k5_square_1(A) ) )).

fof(t54_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_int_1(B)
         => ~ ( A != np__0
              & ? [C] :
                  ( v1_int_1(C)
                  & B = k3_xcmplx_0(np__2,C) )
              & k3_power(k4_xcmplx_0(A),B) != k3_power(A,B) ) ) ) )).

fof(t55_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_int_1(B)
         => ~ ( A != np__0
              & ? [C] :
                  ( v1_int_1(C)
                  & B = k2_xcmplx_0(k3_xcmplx_0(np__2,C),np__1) )
              & k3_power(k4_xcmplx_0(A),B) != k4_xcmplx_0(k3_power(A,B)) ) ) ) )).

fof(t56_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ~ r1_xreal_0(A,k1_real_1(np__1))
           => r1_xreal_0(k2_xcmplx_0(np__1,k3_xcmplx_0(B,A)),k3_power(k2_xcmplx_0(np__1,A),B)) ) ) ) )).

fof(t57_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ~ ( ~ r1_xreal_0(A,np__0)
                  & A != np__1
                  & B != C
                  & k3_power(A,B) = k3_power(A,C) ) ) ) ) )).

fof(d3_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ~ ( ~ r1_xreal_0(A,np__0)
              & A != np__1
              & ~ r1_xreal_0(B,np__0)
              & ~ ! [C] :
                    ( v1_xreal_0(C)
                   => ( C = k5_power(A,B)
                    <=> k3_power(A,C) = B ) ) ) ) ) )).

fof(t58_power,axiom,(
    $true )).

fof(t59_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ~ ( ~ r1_xreal_0(A,np__0)
          & A != np__1
          & k5_power(A,np__1) != np__0 ) ) )).

fof(t60_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ~ ( ~ r1_xreal_0(A,np__0)
          & A != np__1
          & k5_power(A,A) != np__1 ) ) )).

fof(t61_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ~ ( ~ r1_xreal_0(A,np__0)
                  & A != np__1
                  & ~ r1_xreal_0(B,np__0)
                  & ~ r1_xreal_0(C,np__0)
                  & k2_xcmplx_0(k5_power(A,B),k5_power(A,C)) != k5_power(A,k3_xcmplx_0(B,C)) ) ) ) ) )).

fof(t62_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ~ ( ~ r1_xreal_0(A,np__0)
                  & A != np__1
                  & ~ r1_xreal_0(B,np__0)
                  & ~ r1_xreal_0(C,np__0)
                  & k6_xcmplx_0(k5_power(A,B),k5_power(A,C)) != k5_power(A,k7_xcmplx_0(B,C)) ) ) ) ) )).

fof(t63_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ~ ( ~ r1_xreal_0(A,np__0)
                  & A != np__1
                  & ~ r1_xreal_0(B,np__0)
                  & k5_power(A,k3_power(B,C)) != k3_xcmplx_0(C,k5_power(A,B)) ) ) ) ) )).

fof(t64_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ~ ( ~ r1_xreal_0(A,np__0)
                  & A != np__1
                  & ~ r1_xreal_0(B,np__0)
                  & B != np__1
                  & ~ r1_xreal_0(C,np__0)
                  & k5_power(A,C) != k3_xcmplx_0(k5_power(A,B),k5_power(B,C)) ) ) ) ) )).

fof(t65_power,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,np__0)
                  & ~ r1_xreal_0(C,B)
                  & r1_xreal_0(k5_power(A,C),k5_power(A,B)) ) ) ) ) )).

fof(t66_power,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(np__1,A)
                  & ~ r1_xreal_0(B,np__0)
                  & ~ r1_xreal_0(C,B)
                  & r1_xreal_0(k5_power(A,B),k5_power(A,C)) ) ) ) ) )).

fof(t67_power,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,A,B) = k4_power(k3_real_1(np__1,k6_real_1(np__1,k1_nat_1(B,np__1))),k1_nat_1(B,np__1)) )
       => v4_seq_2(A) ) ) )).

fof(d4_power,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( A = k7_power
      <=> ! [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)
                 => k2_seq_1(k5_numbers,k1_numbers,B,C) = k4_power(k3_real_1(np__1,k6_real_1(np__1,k1_nat_1(C,np__1))),k1_nat_1(C,np__1)) )
             => A = k2_seq_2(B) ) ) ) ) )).

fof(dt_k1_power,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & v1_xreal_0(B) )
     => v1_xreal_0(k1_power(A,B)) ) )).

fof(dt_k2_power,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k1_numbers) )
     => m1_subset_1(k2_power(A,B),k1_numbers) ) )).

fof(redefinition_k2_power,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k1_numbers) )
     => k2_power(A,B) = k1_power(A,B) ) )).

fof(dt_k3_power,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => v1_xreal_0(k3_power(A,B)) ) )).

fof(dt_k4_power,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => m1_subset_1(k4_power(A,B),k1_numbers) ) )).

fof(redefinition_k4_power,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => k4_power(A,B) = k3_power(A,B) ) )).

fof(dt_k5_power,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => v1_xreal_0(k5_power(A,B)) ) )).

fof(dt_k6_power,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => m1_subset_1(k6_power(A,B),k1_numbers) ) )).

fof(redefinition_k6_power,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => k6_power(A,B) = k5_power(A,B) ) )).

fof(dt_k7_power,axiom,(
    v1_xreal_0(k7_power) )).

fof(dt_k8_power,axiom,(
    m1_subset_1(k8_power,k1_numbers) )).

fof(redefinition_k8_power,axiom,(
    k8_power = k7_power )).
%------------------------------------------------------------------------------