SET007 Axioms: SET007+831.ax


%------------------------------------------------------------------------------
% File     : SET007+831 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Roots of the Special Polynomial Equation with Real Coefficients
% 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    : polyeq_4 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   30 (   4 unit)
%            Number of atoms       :  312 ( 143 equality)
%            Maximal formula depth :   27 (  14 average)
%            Number of connectives :  379 (  97 ~  ;  21  |; 145  &)
%                                         (   0 <=>; 116 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    5 (   1 propositional; 0-3 arity)
%            Number of functors    :   25 (   8 constant; 0-7 arity)
%            Number of variables   :  118 (   0 singleton; 107 !;  11 ?)
%            Maximal term depth    :    9 (   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_polyeq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( r1_xreal_0(np__0,k2_quin_1(A,B,C))
               => ( A = np__0
                  | r1_xreal_0(np__0,k6_real_1(B,A))
                  | r1_xreal_0(k6_real_1(C,A),np__0)
                  | ( ~ r1_xreal_0(k6_real_1(k3_real_1(k1_real_1(B),k9_square_1(k2_quin_1(A,B,C))),k4_real_1(np__2,A)),np__0)
                    & ~ r1_xreal_0(k6_real_1(k5_real_1(k1_real_1(B),k9_square_1(k2_quin_1(A,B,C))),k4_real_1(np__2,A)),np__0) ) ) ) ) ) ) )).

fof(t2_polyeq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( r1_xreal_0(np__0,k2_quin_1(A,B,C))
               => ( A = np__0
                  | r1_xreal_0(k6_real_1(B,A),np__0)
                  | r1_xreal_0(k6_real_1(C,A),np__0)
                  | ( ~ r1_xreal_0(np__0,k6_real_1(k3_real_1(k1_real_1(B),k9_square_1(k2_quin_1(A,B,C))),k4_real_1(np__2,A)))
                    & ~ r1_xreal_0(np__0,k6_real_1(k5_real_1(k1_real_1(B),k9_square_1(k2_quin_1(A,B,C))),k4_real_1(np__2,A))) ) ) ) ) ) ) )).

fof(t3_polyeq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ~ ( A != np__0
                  & ~ r1_xreal_0(np__0,k6_real_1(B,A))
                  & ~ ( ~ r1_xreal_0(k6_real_1(k3_real_1(k1_real_1(C),k9_square_1(k2_quin_1(A,C,B))),k4_real_1(np__2,A)),np__0)
                      & ~ r1_xreal_0(np__0,k6_real_1(k5_real_1(k1_real_1(C),k9_square_1(k2_quin_1(A,C,B))),k4_real_1(np__2,A))) )
                  & ~ ( ~ r1_xreal_0(np__0,k6_real_1(k3_real_1(k1_real_1(C),k9_square_1(k2_quin_1(A,C,B))),k4_real_1(np__2,A)))
                      & ~ r1_xreal_0(k6_real_1(k5_real_1(k1_real_1(C),k9_square_1(k2_quin_1(A,C,B))),k4_real_1(np__2,A)),np__0) ) ) ) ) ) )).

fof(t4_polyeq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ~ ( ~ r1_xreal_0(A,np__0)
                  & ? [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                      & C = k2_nat_1(np__2,D)
                      & r1_xreal_0(np__1,D) )
                  & k3_newton(B,C) = A
                  & B != k2_power(C,A)
                  & B != k1_real_1(k2_power(C,A)) ) ) ) ) )).

fof(t5_polyeq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ~ ( A != np__0
                  & k4_polyeq_1(A,B,np__0,C) = np__0
                  & C != np__0
                  & C != k1_real_1(k6_real_1(B,A)) ) ) ) ) )).

fof(t6_polyeq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( k4_polyeq_1(A,np__0,np__0,B) = np__0
           => ( A = np__0
              | B = np__0 ) ) ) ) )).

fof(t7_polyeq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ~ ( A != np__0
                          & ? [F] :
                              ( m2_subset_1(F,k1_numbers,k5_numbers)
                              & E = k1_nat_1(k2_nat_1(np__2,F),np__1) )
                          & r1_xreal_0(np__0,k2_quin_1(A,B,C))
                          & k4_polyeq_1(A,B,C,k3_newton(D,E)) = np__0
                          & D != k2_power(E,k6_real_1(k3_real_1(k1_real_1(B),k9_square_1(k2_quin_1(A,B,C))),k4_real_1(np__2,A)))
                          & D != k2_power(E,k6_real_1(k5_real_1(k1_real_1(B),k9_square_1(k2_quin_1(A,B,C))),k4_real_1(np__2,A))) ) ) ) ) ) ) )).

fof(t8_polyeq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ~ ( A != np__0
                          & ~ r1_xreal_0(np__0,k6_real_1(B,A))
                          & ~ r1_xreal_0(k6_real_1(C,A),np__0)
                          & ? [F] :
                              ( m2_subset_1(F,k1_numbers,k5_numbers)
                              & E = k2_nat_1(np__2,F)
                              & r1_xreal_0(np__1,F) )
                          & r1_xreal_0(np__0,k2_quin_1(A,B,C))
                          & k4_polyeq_1(A,B,C,k3_newton(D,E)) = np__0
                          & D != k2_power(E,k6_real_1(k3_real_1(k1_real_1(B),k9_square_1(k2_quin_1(A,B,C))),k4_real_1(np__2,A)))
                          & D != k1_real_1(k2_power(E,k6_real_1(k3_real_1(k1_real_1(B),k9_square_1(k2_quin_1(A,B,C))),k4_real_1(np__2,A))))
                          & D != k2_power(E,k6_real_1(k5_real_1(k1_real_1(B),k9_square_1(k2_quin_1(A,B,C))),k4_real_1(np__2,A)))
                          & D != k1_real_1(k2_power(E,k6_real_1(k5_real_1(k1_real_1(B),k9_square_1(k2_quin_1(A,B,C))),k4_real_1(np__2,A)))) ) ) ) ) ) ) )).

fof(t9_polyeq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ~ ( A != np__0
                      & ? [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                          & D = k1_nat_1(k2_nat_1(np__2,E),np__1) )
                      & k4_polyeq_1(A,B,np__0,k3_newton(C,D)) = np__0
                      & C != np__0
                      & C != k2_power(D,k1_real_1(k6_real_1(B,A))) ) ) ) ) ) )).

fof(t10_polyeq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ~ ( A != np__0
                      & ~ r1_xreal_0(np__0,k6_real_1(B,A))
                      & ? [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                          & D = k2_nat_1(np__2,E)
                          & r1_xreal_0(np__1,E) )
                      & k4_polyeq_1(A,B,np__0,k3_newton(C,D)) = np__0
                      & C != np__0
                      & C != k2_power(D,k1_real_1(k6_real_1(B,A)))
                      & C != k1_real_1(k2_power(D,k1_real_1(k6_real_1(B,A)))) ) ) ) ) ) )).

fof(t11_polyeq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( k3_real_1(k3_newton(A,np__3),k3_newton(B,np__3)) = k4_real_1(k3_real_1(A,B),k3_real_1(k5_real_1(k7_square_1(A),k4_real_1(A,B)),k7_square_1(B)))
            & k3_real_1(k3_newton(A,np__5),k3_newton(B,np__5)) = k4_real_1(k3_real_1(A,B),k3_real_1(k5_real_1(k3_real_1(k5_real_1(k3_newton(A,np__4),k4_real_1(k3_newton(A,np__3),B)),k4_real_1(k3_newton(A,np__2),k3_newton(B,np__2))),k4_real_1(A,k3_newton(B,np__3))),k3_newton(B,np__4))) ) ) ) )).

fof(t12_polyeq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ~ ( A != np__0
                  & r1_xreal_0(np__0,k5_real_1(k5_real_1(k7_square_1(B),k4_real_1(k4_real_1(np__2,A),B)),k4_real_1(np__3,k7_square_1(A))))
                  & k8_polyeq_1(A,B,B,A,C) = np__0
                  & C != k1_real_1(np__1)
                  & C != k6_real_1(k3_real_1(k5_real_1(A,B),k9_square_1(k5_real_1(k5_real_1(k7_square_1(B),k4_real_1(k4_real_1(np__2,A),B)),k4_real_1(np__3,k7_square_1(A))))),k4_real_1(np__2,A))
                  & C != k6_real_1(k5_real_1(k5_real_1(A,B),k9_square_1(k5_real_1(k5_real_1(k7_square_1(B),k4_real_1(k4_real_1(np__2,A),B)),k4_real_1(np__3,k7_square_1(A))))),k4_real_1(np__2,A)) ) ) ) ) )).

fof(d1_polyeq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] :
                      ( m1_subset_1(E,k1_numbers)
                     => ! [F] :
                          ( m1_subset_1(F,k1_numbers)
                         => ! [G] :
                              ( m1_subset_1(G,k1_numbers)
                             => k1_polyeq_4(A,B,C,D,E,F,G) = k3_real_1(k3_real_1(k3_real_1(k3_real_1(k3_real_1(k4_real_1(A,k3_newton(G,np__5)),k4_real_1(B,k3_newton(G,np__4))),k4_real_1(C,k3_newton(G,np__3))),k4_real_1(D,k7_square_1(G))),k4_real_1(E,G)),F) ) ) ) ) ) ) ) )).

fof(t13_polyeq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( k1_polyeq_4(A,B,C,C,B,A,D) = np__0
                   => ( A = np__0
                      | r1_xreal_0(k5_real_1(k3_real_1(k3_real_1(k7_square_1(B),k4_real_1(k4_real_1(np__2,A),B)),k4_real_1(np__5,k7_square_1(A))),k4_real_1(k4_real_1(np__4,A),C)),np__0)
                      | ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ! [F] :
                              ( m1_subset_1(F,k1_numbers)
                             => ~ ( E = k6_real_1(k3_real_1(k5_real_1(A,B),k9_square_1(k5_real_1(k3_real_1(k3_real_1(k7_square_1(B),k4_real_1(k4_real_1(np__2,A),B)),k4_real_1(np__5,k7_square_1(A))),k4_real_1(k4_real_1(np__4,A),C)))),k4_real_1(np__2,A))
                                  & F = k6_real_1(k5_real_1(k5_real_1(A,B),k9_square_1(k5_real_1(k3_real_1(k3_real_1(k7_square_1(B),k4_real_1(k4_real_1(np__2,A),B)),k4_real_1(np__5,k7_square_1(A))),k4_real_1(k4_real_1(np__4,A),C)))),k4_real_1(np__2,A))
                                  & D != k1_real_1(np__1)
                                  & D != k6_real_1(k3_real_1(E,k9_square_1(k2_quin_1(np__1,k1_real_1(E),np__1))),np__2)
                                  & D != k6_real_1(k3_real_1(F,k9_square_1(k2_quin_1(np__1,k1_real_1(F),np__1))),np__2)
                                  & D != k6_real_1(k5_real_1(E,k9_square_1(k2_quin_1(np__1,k1_real_1(E),np__1))),np__2)
                                  & D != k6_real_1(k5_real_1(F,k9_square_1(k2_quin_1(np__1,k1_real_1(F),np__1))),np__2) ) ) ) ) ) ) ) ) ) )).

fof(t14_polyeq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ~ ( k3_real_1(A,B) = C
                      & k4_real_1(A,B) = D
                      & r1_xreal_0(np__0,k5_real_1(k7_square_1(C),k4_real_1(np__4,D)))
                      & ~ ( A = k6_real_1(k3_real_1(C,k9_square_1(k5_real_1(k7_square_1(C),k4_real_1(np__4,D)))),np__2)
                          & B = k6_real_1(k5_real_1(C,k9_square_1(k5_real_1(k7_square_1(C),k4_real_1(np__4,D)))),np__2) )
                      & ~ ( A = k6_real_1(k5_real_1(C,k9_square_1(k5_real_1(k7_square_1(C),k4_real_1(np__4,D)))),np__2)
                          & B = k6_real_1(k3_real_1(C,k9_square_1(k5_real_1(k7_square_1(C),k4_real_1(np__4,D)))),np__2) ) ) ) ) ) ) )).

fof(t15_polyeq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ~ ( k3_real_1(k3_newton(A,E),k3_newton(B,E)) = C
                          & k4_real_1(k3_newton(A,E),k3_newton(B,E)) = D
                          & r1_xreal_0(np__0,k5_real_1(k7_square_1(C),k4_real_1(np__4,D)))
                          & ? [F] :
                              ( m2_subset_1(F,k1_numbers,k5_numbers)
                              & E = k1_nat_1(k2_nat_1(np__2,F),np__1) )
                          & ~ ( A = k2_power(E,k6_real_1(k3_real_1(C,k9_square_1(k5_real_1(k7_square_1(C),k4_real_1(np__4,D)))),np__2))
                              & B = k2_power(E,k6_real_1(k5_real_1(C,k9_square_1(k5_real_1(k7_square_1(C),k4_real_1(np__4,D)))),np__2)) )
                          & ~ ( A = k2_power(E,k6_real_1(k5_real_1(C,k9_square_1(k5_real_1(k7_square_1(C),k4_real_1(np__4,D)))),np__2))
                              & B = k2_power(E,k6_real_1(k3_real_1(C,k9_square_1(k5_real_1(k7_square_1(C),k4_real_1(np__4,D)))),np__2)) ) ) ) ) ) ) ) )).

fof(t16_polyeq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ~ ( k3_real_1(k3_newton(A,E),k3_newton(B,E)) = C
                          & k4_real_1(k3_newton(A,E),k3_newton(B,E)) = D
                          & r1_xreal_0(np__0,k5_real_1(k7_square_1(C),k4_real_1(np__4,D)))
                          & ~ r1_xreal_0(C,np__0)
                          & ~ r1_xreal_0(D,np__0)
                          & ? [F] :
                              ( m2_subset_1(F,k1_numbers,k5_numbers)
                              & E = k2_nat_1(np__2,F)
                              & r1_xreal_0(np__1,F) )
                          & ~ ( A = k2_power(E,k6_real_1(k3_real_1(C,k9_square_1(k5_real_1(k7_square_1(C),k4_real_1(np__4,D)))),np__2))
                              & B = k2_power(E,k6_real_1(k5_real_1(C,k9_square_1(k5_real_1(k7_square_1(C),k4_real_1(np__4,D)))),np__2)) )
                          & ~ ( A = k1_real_1(k2_power(E,k6_real_1(k3_real_1(C,k9_square_1(k5_real_1(k7_square_1(C),k4_real_1(np__4,D)))),np__2)))
                              & B = k2_power(E,k6_real_1(k5_real_1(C,k9_square_1(k5_real_1(k7_square_1(C),k4_real_1(np__4,D)))),np__2)) )
                          & ~ ( A = k2_power(E,k6_real_1(k3_real_1(C,k9_square_1(k5_real_1(k7_square_1(C),k4_real_1(np__4,D)))),np__2))
                              & B = k1_real_1(k2_power(E,k6_real_1(k5_real_1(C,k9_square_1(k5_real_1(k7_square_1(C),k4_real_1(np__4,D)))),np__2))) )
                          & ~ ( A = k1_real_1(k2_power(E,k6_real_1(k3_real_1(C,k9_square_1(k5_real_1(k7_square_1(C),k4_real_1(np__4,D)))),np__2)))
                              & B = k1_real_1(k2_power(E,k6_real_1(k5_real_1(C,k9_square_1(k5_real_1(k7_square_1(C),k4_real_1(np__4,D)))),np__2))) )
                          & ~ ( A = k2_power(E,k6_real_1(k5_real_1(C,k9_square_1(k5_real_1(k7_square_1(C),k4_real_1(np__4,D)))),np__2))
                              & B = k2_power(E,k6_real_1(k3_real_1(C,k9_square_1(k5_real_1(k7_square_1(C),k4_real_1(np__4,D)))),np__2)) )
                          & ~ ( A = k1_real_1(k2_power(E,k6_real_1(k5_real_1(C,k9_square_1(k5_real_1(k7_square_1(C),k4_real_1(np__4,D)))),np__2)))
                              & B = k2_power(E,k6_real_1(k3_real_1(C,k9_square_1(k5_real_1(k7_square_1(C),k4_real_1(np__4,D)))),np__2)) )
                          & ~ ( A = k2_power(E,k6_real_1(k5_real_1(C,k9_square_1(k5_real_1(k7_square_1(C),k4_real_1(np__4,D)))),np__2))
                              & B = k1_real_1(k2_power(E,k6_real_1(k3_real_1(C,k9_square_1(k5_real_1(k7_square_1(C),k4_real_1(np__4,D)))),np__2))) )
                          & ~ ( A = k1_real_1(k2_power(E,k6_real_1(k5_real_1(C,k9_square_1(k5_real_1(k7_square_1(C),k4_real_1(np__4,D)))),np__2)))
                              & B = k1_real_1(k2_power(E,k6_real_1(k3_real_1(C,k9_square_1(k5_real_1(k7_square_1(C),k4_real_1(np__4,D)))),np__2))) ) ) ) ) ) ) ) )).

fof(t17_polyeq_4,axiom,(
    $true )).

fof(t18_polyeq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ~ ( k3_real_1(k3_newton(A,E),k3_newton(B,E)) = C
                          & k5_real_1(k3_newton(A,E),k3_newton(B,E)) = D
                          & ? [F] :
                              ( m2_subset_1(F,k1_numbers,k5_numbers)
                              & E = k2_nat_1(np__2,F)
                              & r1_xreal_0(np__1,F) )
                          & ~ r1_xreal_0(C,np__0)
                          & ~ r1_xreal_0(k3_real_1(C,D),np__0)
                          & ~ r1_xreal_0(k5_real_1(C,D),np__0)
                          & ~ ( A = k2_power(E,k6_real_1(k3_real_1(C,D),np__2))
                              & B = k2_power(E,k6_real_1(k5_real_1(C,D),np__2)) )
                          & ~ ( A = k2_power(E,k6_real_1(k3_real_1(C,D),np__2))
                              & B = k1_real_1(k2_power(E,k6_real_1(k5_real_1(C,D),np__2))) )
                          & ~ ( A = k1_real_1(k2_power(E,k6_real_1(k3_real_1(C,D),np__2)))
                              & B = k2_power(E,k6_real_1(k5_real_1(C,D),np__2)) )
                          & ~ ( A = k1_real_1(k2_power(E,k6_real_1(k3_real_1(C,D),np__2)))
                              & B = k1_real_1(k2_power(E,k6_real_1(k5_real_1(C,D),np__2))) ) ) ) ) ) ) ) )).

fof(t19_polyeq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] :
                      ( m1_subset_1(E,k1_numbers)
                     => ! [F] :
                          ( m2_subset_1(F,k1_numbers,k5_numbers)
                         => ~ ( k3_real_1(k4_real_1(A,k3_newton(B,F)),k4_real_1(C,k3_newton(D,F))) = E
                              & k4_real_1(B,D) = np__0
                              & ? [G] :
                                  ( m2_subset_1(G,k1_numbers,k5_numbers)
                                  & F = k1_nat_1(k2_nat_1(np__2,G),np__1) )
                              & k4_real_1(A,C) != np__0
                              & ~ ( B = np__0
                                  & D = k2_power(F,k6_real_1(E,C)) )
                              & ~ ( B = k2_power(F,k6_real_1(E,A))
                                  & D = np__0 ) ) ) ) ) ) ) ) )).

fof(t20_polyeq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] :
                      ( m1_subset_1(E,k1_numbers)
                     => ! [F] :
                          ( m2_subset_1(F,k1_numbers,k5_numbers)
                         => ~ ( k3_real_1(k4_real_1(A,k3_newton(B,F)),k4_real_1(C,k3_newton(D,F))) = E
                              & k4_real_1(B,D) = np__0
                              & ? [G] :
                                  ( m2_subset_1(G,k1_numbers,k5_numbers)
                                  & F = k2_nat_1(np__2,G)
                                  & r1_xreal_0(np__1,G) )
                              & ~ r1_xreal_0(k6_real_1(E,C),np__0)
                              & ~ r1_xreal_0(k6_real_1(E,A),np__0)
                              & k4_real_1(A,C) != np__0
                              & ~ ( B = np__0
                                  & D = k2_power(F,k6_real_1(E,C)) )
                              & ~ ( B = np__0
                                  & D = k1_real_1(k2_power(F,k6_real_1(E,C))) )
                              & ~ ( B = k2_power(F,k6_real_1(E,A))
                                  & D = np__0 )
                              & ~ ( B = k1_real_1(k2_power(F,k6_real_1(E,A)))
                                  & D = np__0 ) ) ) ) ) ) ) ) )).

fof(t21_polyeq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] :
                      ( m1_subset_1(E,k1_numbers)
                     => ! [F] :
                          ( m2_subset_1(F,k1_numbers,k5_numbers)
                         => ( ( k4_real_1(A,k3_newton(B,F)) = C
                              & k4_real_1(B,D) = E )
                           => ( ! [G] :
                                  ( m2_subset_1(G,k1_numbers,k5_numbers)
                                 => F != k1_nat_1(k2_nat_1(np__2,G),np__1) )
                              | k4_real_1(C,A) = np__0
                              | ( B = k2_power(F,k6_real_1(C,A))
                                & D = k4_real_1(E,k2_power(F,k6_real_1(A,C))) ) ) ) ) ) ) ) ) ) )).

fof(t22_polyeq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] :
                      ( m1_subset_1(E,k1_numbers)
                     => ! [F] :
                          ( m2_subset_1(F,k1_numbers,k5_numbers)
                         => ~ ( k4_real_1(A,k3_newton(B,F)) = C
                              & k4_real_1(B,D) = E
                              & ? [G] :
                                  ( m2_subset_1(G,k1_numbers,k5_numbers)
                                  & F = k2_nat_1(np__2,G)
                                  & r1_xreal_0(np__1,G) )
                              & ~ r1_xreal_0(k6_real_1(C,A),np__0)
                              & A != np__0
                              & ~ ( B = k2_power(F,k6_real_1(C,A))
                                  & D = k4_real_1(E,k2_power(F,k6_real_1(A,C))) )
                              & ~ ( B = k1_real_1(k2_power(F,k6_real_1(C,A)))
                                  & D = k1_real_1(k4_real_1(E,k2_power(F,k6_real_1(A,C)))) ) ) ) ) ) ) ) ) )).

fof(t23_polyeq_4,axiom,(
    $true )).

fof(t24_polyeq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( k4_power(A,B) = np__1
           => ( r1_xreal_0(A,np__0)
              | A = np__1
              | B = np__0 ) ) ) ) )).

fof(t25_polyeq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( k4_power(A,B) = A
           => ( r1_xreal_0(A,np__0)
              | A = np__1
              | B = np__1 ) ) ) ) )).

fof(t26_polyeq_4,axiom,(
    $true )).

fof(t27_polyeq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( k6_power(A,C) = np__0
               => ( r1_xreal_0(A,np__0)
                  | A = np__1
                  | r1_xreal_0(C,np__0)
                  | C = np__1 ) ) ) ) ) )).

fof(t28_polyeq_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( k6_power(A,C) = np__1
               => ( r1_xreal_0(A,np__0)
                  | A = np__1
                  | r1_xreal_0(C,np__0)
                  | C = A ) ) ) ) ) )).

fof(dt_k1_polyeq_4,axiom,(
    $true )).
%------------------------------------------------------------------------------