SET007 Axioms: SET007+197.ax


%------------------------------------------------------------------------------
% File     : SET007+197 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Chinese Remainder Theorem
% 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    : wsierp_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   75 (   6 unit)
%            Number of atoms       :  384 (  90 equality)
%            Maximal formula depth :   24 (   8 average)
%            Number of connectives :  357 (  48 ~  ;   9  |; 110  &)
%                                         (   6 <=>; 184 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   21 (   1 propositional; 0-3 arity)
%            Number of functors    :   51 (   9 constant; 0-4 arity)
%            Number of variables   :  191 (   0 singleton; 183 !;   8 ?)
%            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(fc1_wsierp_1,axiom,(
    ! [A,B] :
      ( ( v1_int_1(A)
        & m1_subset_1(B,k5_numbers) )
     => ( v1_xcmplx_0(k2_newton(A,B))
        & v1_xreal_0(k2_newton(A,B))
        & v1_int_1(k2_newton(A,B))
        & v1_rat_1(k2_newton(A,B)) ) ) )).

fof(fc2_wsierp_1,axiom,(
    ! [A,B] :
      ( m1_finseq_1(A,k4_numbers)
     => ( v1_xcmplx_0(k1_funct_1(A,B))
        & v1_xreal_0(k1_funct_1(A,B))
        & v1_int_1(k1_funct_1(A,B))
        & v1_rat_1(k1_funct_1(A,B)) ) ) )).

fof(t1_wsierp_1,axiom,(
    $true )).

fof(t2_wsierp_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( k2_newton(A,np__2) = k3_xcmplx_0(A,A)
        & k2_newton(k4_xcmplx_0(A),np__2) = k2_newton(A,np__2) ) ) )).

fof(t3_wsierp_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( k2_newton(k4_xcmplx_0(A),k2_nat_1(np__2,B)) = k2_newton(A,k2_nat_1(np__2,B))
            & k2_newton(k4_xcmplx_0(A),k1_nat_1(k2_nat_1(np__2,B),np__1)) = k4_xcmplx_0(k2_newton(A,k1_nat_1(k2_nat_1(np__2,B),np__1))) ) ) ) )).

fof(t4_wsierp_1,axiom,(
    $true )).

fof(t5_wsierp_1,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)
                  & k2_newton(A,C) = k2_newton(B,C) )
               => ( r1_xreal_0(C,np__0)
                  | A = B ) ) ) ) ) )).

fof(t6_wsierp_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( ~ r1_xreal_0(A,k2_square_1(B,C))
              <=> ( ~ r1_xreal_0(A,B)
                  & ~ r1_xreal_0(A,C) ) ) ) ) ) )).

fof(t7_wsierp_1,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(C,B) )
               => ( r1_xreal_0(C,k6_xcmplx_0(B,A))
                  & r1_xreal_0(k2_xcmplx_0(C,A),B) ) ) ) ) ) )).

fof(t8_wsierp_1,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,C) )
                  | ( ~ r1_xreal_0(np__0,A)
                    & r1_xreal_0(C,B) ) )
               => ( ~ r1_xreal_0(B,k2_xcmplx_0(C,A))
                  & ~ r1_xreal_0(k6_xcmplx_0(B,A),C) ) ) ) ) ) )).

fof(t9_wsierp_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ( ( r2_int_1(A,B)
                  & r2_int_1(A,C) )
               => r2_int_1(A,k2_xcmplx_0(B,C)) ) ) ) ) )).

fof(t10_wsierp_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ! [D] :
                  ( v1_int_1(D)
                 => ! [E] :
                      ( v1_int_1(E)
                     => ( ( r2_int_1(A,B)
                          & r2_int_1(A,C) )
                       => r2_int_1(A,k2_xcmplx_0(k3_xcmplx_0(B,D),k3_xcmplx_0(C,E))) ) ) ) ) ) ) )).

fof(t11_wsierp_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ( ( k3_int_2(A,B) = np__1
                  & k3_int_2(C,B) = np__1 )
               => k3_int_2(k3_xcmplx_0(A,C),B) = np__1 ) ) ) ) )).

fof(t12_wsierp_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( k6_nat_1(A,B) = np__1
                  & k6_nat_1(C,B) = np__1 )
               => k6_nat_1(k2_nat_1(A,C),B) = np__1 ) ) ) ) )).

fof(t13_wsierp_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ( k3_int_2(np__0,A) = k1_prepower(A)
        & k3_int_2(np__1,A) = np__1 ) ) )).

fof(t14_wsierp_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => r1_int_2(np__1,A) ) )).

fof(t15_wsierp_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ( r1_int_2(B,C)
               => r1_int_2(k2_newton(B,A),C) ) ) ) ) )).

fof(t16_wsierp_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( v1_int_1(C)
             => ! [D] :
                  ( v1_int_1(D)
                 => ( r1_int_2(C,D)
                   => r1_int_2(k2_newton(C,A),k2_newton(D,B)) ) ) ) ) ) )).

fof(t17_wsierp_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( v1_int_1(C)
             => ! [D] :
                  ( v1_int_1(D)
                 => ( k3_int_2(C,D) = np__1
                   => ( k3_int_2(C,k2_newton(D,A)) = np__1
                      & k3_int_2(k2_newton(C,B),k2_newton(D,A)) = np__1 ) ) ) ) ) ) )).

fof(t18_wsierp_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ( r2_int_1(k1_prepower(A),B)
          <=> r2_int_1(A,B) ) ) ) )).

fof(t19_wsierp_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r1_nat_1(A,B)
               => r1_nat_1(k2_wsierp_1(A,C),k2_wsierp_1(B,C)) ) ) ) ) )).

fof(t20_wsierp_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( r1_nat_1(A,np__1)
       => A = np__1 ) ) )).

fof(t21_wsierp_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( r1_nat_1(A,B)
                  & k6_nat_1(B,C) = np__1 )
               => k6_nat_1(A,C) = np__1 ) ) ) ) )).

fof(t22_wsierp_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ( A != np__0
           => ( r2_int_1(A,B)
            <=> v1_int_1(k7_xcmplx_0(B,A)) ) ) ) ) )).

fof(t23_wsierp_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r1_xreal_0(A,k5_real_1(B,C))
               => ( r1_xreal_0(A,B)
                  & r1_xreal_0(C,B) ) ) ) ) ) )).

fof(d1_wsierp_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C) )
             => ( ( ~ r2_hidden(A,k4_finseq_1(B))
                 => ( C = k2_finseq_3(A,B)
                  <=> C = B ) )
                & ( r2_hidden(A,k4_finseq_1(B))
                 => ( C = k2_finseq_3(A,B)
                  <=> ( k1_nat_1(k3_finseq_1(C),np__1) = k3_finseq_1(B)
                      & ! [D] :
                          ( m2_subset_1(D,k1_numbers,k5_numbers)
                         => ( ( ~ r1_xreal_0(A,D)
                             => k1_funct_1(C,D) = k1_funct_1(B,D) )
                            & ( r1_xreal_0(A,D)
                             => k1_funct_1(C,D) = k1_funct_1(B,k1_nat_1(D,np__1)) ) ) ) ) ) ) ) ) ) ) )).

fof(t24_wsierp_1,axiom,(
    $true )).

fof(t25_wsierp_1,axiom,(
    $true )).

fof(t26_wsierp_1,axiom,(
    ! [A,B,C] :
      ( k2_finseq_3(np__1,k9_finseq_1(A)) = k1_xboole_0
      & k2_finseq_3(np__1,k10_finseq_1(A,B)) = k9_finseq_1(B)
      & k2_finseq_3(np__2,k10_finseq_1(A,B)) = k9_finseq_1(A)
      & k2_finseq_3(np__1,k11_finseq_1(A,B,C)) = k10_finseq_1(B,C)
      & k2_finseq_3(np__2,k11_finseq_1(A,B,C)) = k10_finseq_1(A,C)
      & k2_finseq_3(np__3,k11_finseq_1(A,B,C)) = k10_finseq_1(A,B) ) )).

fof(t27_wsierp_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => ( r2_hidden(A,k4_finseq_1(B))
           => k3_real_1(k15_rvsum_1(k11_wsierp_1(k1_numbers,A,B)),k1_wsierp_1(B,A)) = k15_rvsum_1(B) ) ) ) )).

fof(t28_wsierp_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_trees_4(B,k1_numbers,k5_numbers)
         => ( r2_hidden(A,k4_finseq_1(B))
           => m2_subset_1(k6_real_1(k10_wsierp_1(B),k3_wsierp_1(B,A)),k1_numbers,k5_numbers) ) ) ) )).

fof(t29_wsierp_1,axiom,(
    ! [A] :
      ( v1_rat_1(A)
     => r1_int_2(k2_rat_1(A),k1_rat_1(A)) ) )).

fof(t30_wsierp_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_rat_1(C)
             => ( ( C = k7_xcmplx_0(B,A)
                  & r1_int_2(B,A) )
               => ( C = np__0
                  | A = np__0
                  | ( B = k2_rat_1(C)
                    & A = k1_rat_1(C) ) ) ) ) ) ) )).

fof(t31_wsierp_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ~ ( ? [C] :
                  ( v1_rat_1(C)
                  & A = k2_newton(C,B) )
              & ! [C] :
                  ( v1_int_1(C)
                 => A != k2_newton(C,B) ) ) ) ) )).

fof(t32_wsierp_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ~ ( ? [C] :
                  ( v1_rat_1(C)
                  & A = k2_newton(C,B) )
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => A != k2_wsierp_1(C,B) ) ) ) ) )).

fof(t33_wsierp_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r1_nat_1(k2_wsierp_1(B,A),k2_wsierp_1(C,A))
               => ( r1_xreal_0(A,np__0)
                  | r1_nat_1(B,C) ) ) ) ) ) )).

fof(t34_wsierp_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ? [C] :
              ( v1_int_1(C)
              & ? [D] :
                  ( v1_int_1(D)
                  & k6_nat_1(A,B) = k2_xcmplx_0(k3_xcmplx_0(A,C),k3_xcmplx_0(B,D)) ) ) ) ) )).

fof(t35_wsierp_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ? [C] :
              ( v1_int_1(C)
              & ? [D] :
                  ( v1_int_1(D)
                  & k3_int_2(A,B) = k2_xcmplx_0(k3_xcmplx_0(A,C),k3_xcmplx_0(B,D)) ) ) ) ) )).

fof(t36_wsierp_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ( ( r2_int_1(A,k3_xcmplx_0(B,C))
                  & k3_int_2(A,B) = np__1 )
               => r2_int_1(A,C) ) ) ) ) )).

fof(t37_wsierp_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( k6_nat_1(A,B) = np__1
                  & r1_nat_1(A,k2_nat_1(B,C)) )
               => r1_nat_1(A,C) ) ) ) ) )).

fof(t38_wsierp_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ~ ( A != np__0
              & B != np__0
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => k6_nat_1(A,B) != k5_real_1(k2_nat_1(A,C),k2_nat_1(B,D)) ) ) ) ) ) )).

fof(t39_wsierp_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ~ ( ~ r1_xreal_0(A,np__0)
                      & ~ r1_xreal_0(B,np__0)
                      & k6_nat_1(A,B) = np__1
                      & k2_wsierp_1(C,A) = k2_wsierp_1(D,B)
                      & ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ~ ( C = k2_wsierp_1(E,B)
                              & D = k2_wsierp_1(E,A) ) ) ) ) ) ) ) )).

fof(t40_wsierp_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ( ? [D] :
                    ( v1_int_1(D)
                    & ? [E] :
                        ( v1_int_1(E)
                        & k2_xcmplx_0(k3_xcmplx_0(A,D),k3_xcmplx_0(B,E)) = C ) )
              <=> r2_int_1(k3_int_2(A,B),C) ) ) ) ) )).

fof(t41_wsierp_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ! [D] :
                  ( v1_int_1(D)
                 => ! [E] :
                      ( v1_int_1(E)
                     => ( k2_xcmplx_0(k3_xcmplx_0(A,C),k3_xcmplx_0(B,D)) = E
                       => ( A = np__0
                          | B = np__0
                          | ! [F] :
                              ( v1_int_1(F)
                             => ! [G] :
                                  ( v1_int_1(G)
                                 => ~ ( k2_xcmplx_0(k3_xcmplx_0(A,F),k3_xcmplx_0(B,G)) = E
                                      & ! [H] :
                                          ( v1_int_1(H)
                                         => ~ ( F = k2_xcmplx_0(C,k3_xcmplx_0(H,k7_xcmplx_0(B,k3_int_2(A,B))))
                                              & G = k6_xcmplx_0(D,k3_xcmplx_0(H,k7_xcmplx_0(A,k3_int_2(A,B)))) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t42_wsierp_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ~ ( k6_nat_1(A,B) = np__1
                      & k2_nat_1(A,B) = k2_wsierp_1(C,D)
                      & ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ! [F] :
                              ( m2_subset_1(F,k1_numbers,k5_numbers)
                             => ~ ( A = k2_wsierp_1(E,D)
                                  & B = k2_wsierp_1(F,D) ) ) ) ) ) ) ) ) )).

fof(t43_wsierp_1,axiom,(
    ! [A] :
      ( m1_trees_4(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( r2_hidden(C,k4_finseq_1(A))
                 => k6_nat_1(k3_wsierp_1(A,C),B) = np__1 ) )
           => k6_nat_1(k10_wsierp_1(A),B) = np__1 ) ) ) )).

fof(t44_wsierp_1,axiom,(
    ! [A] :
      ( m1_trees_4(A,k1_numbers,k5_numbers)
     => ( ( r1_xreal_0(np__2,k3_finseq_1(A))
          & ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( ( r2_hidden(B,k4_finseq_1(A))
                      & r2_hidden(C,k4_finseq_1(A)) )
                   => ( B = C
                      | k6_nat_1(k3_wsierp_1(A,B),k3_wsierp_1(A,C)) = np__1 ) ) ) ) )
       => ! [B] :
            ( m1_trees_4(B,k1_numbers,k6_wsierp_1)
           => ~ ( k3_finseq_1(B) = k3_finseq_1(A)
                & ! [C] :
                    ( m1_trees_4(C,k1_numbers,k6_wsierp_1)
                   => ~ ( k3_finseq_1(C) = k3_finseq_1(A)
                        & ! [D] :
                            ( m2_subset_1(D,k1_numbers,k5_numbers)
                           => ( r2_hidden(D,k4_finseq_1(A))
                             => k3_real_1(k4_real_1(k3_wsierp_1(A,D),k1_wsierp_1(C,D)),k1_wsierp_1(B,D)) = k3_real_1(k4_real_1(k3_wsierp_1(A,np__1),k1_wsierp_1(C,np__1)),k1_wsierp_1(B,np__1)) ) ) ) ) ) ) ) ) )).

fof(t45_wsierp_1,axiom,(
    $true )).

fof(t46_wsierp_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( v1_int_1(B)
         => ~ ( A != np__0
              & k3_int_2(A,B) = np__1
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ~ ( np__0 != C
                          & np__0 != D
                          & r1_xreal_0(C,k9_square_1(A))
                          & r1_xreal_0(D,k9_square_1(A))
                          & ( r2_int_1(A,k2_xcmplx_0(k3_xcmplx_0(B,C),D))
                            | r2_int_1(A,k6_xcmplx_0(k3_xcmplx_0(B,C),D)) ) ) ) ) ) ) ) )).

fof(t47_wsierp_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => r1_tarski(k4_finseq_1(k2_finseq_3(A,B)),k4_finseq_1(B)) ) ) )).

fof(t48_wsierp_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( k2_finseq_3(np__1,k7_finseq_1(k9_finseq_1(B),A)) = A
          & k2_finseq_3(k1_nat_1(k3_finseq_1(A),np__1),k7_finseq_1(A,k9_finseq_1(B))) = A ) ) )).

fof(dt_k1_wsierp_1,axiom,(
    ! [A,B] :
      ( ( m1_finseq_1(A,k1_numbers)
        & m1_subset_1(B,k5_numbers) )
     => m1_subset_1(k1_wsierp_1(A,B),k1_numbers) ) )).

fof(redefinition_k1_wsierp_1,axiom,(
    ! [A,B] :
      ( ( m1_finseq_1(A,k1_numbers)
        & m1_subset_1(B,k5_numbers) )
     => k1_wsierp_1(A,B) = k1_funct_1(A,B) ) )).

fof(dt_k2_wsierp_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => m2_subset_1(k2_wsierp_1(A,B),k1_numbers,k5_numbers) ) )).

fof(redefinition_k2_wsierp_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => k2_wsierp_1(A,B) = k2_newton(A,B) ) )).

fof(dt_k3_wsierp_1,axiom,(
    ! [A,B] :
      ( ( m1_finseq_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => m2_subset_1(k3_wsierp_1(A,B),k1_numbers,k5_numbers) ) )).

fof(redefinition_k3_wsierp_1,axiom,(
    ! [A,B] :
      ( ( m1_finseq_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => k3_wsierp_1(A,B) = k1_funct_1(A,B) ) )).

fof(dt_k4_wsierp_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(A))
        & m1_finseq_1(C,B)
        & m1_finseq_1(D,B) )
     => m1_trees_4(k4_wsierp_1(A,B,C,D),A,B) ) )).

fof(redefinition_k4_wsierp_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(A))
        & m1_finseq_1(C,B)
        & m1_finseq_1(D,B) )
     => k4_wsierp_1(A,B,C,D) = k7_finseq_1(C,D) ) )).

fof(dt_k5_wsierp_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ( v1_xboole_0(k5_wsierp_1(A,B))
        & m1_trees_4(k5_wsierp_1(A,B),A,B) ) ) )).

fof(redefinition_k5_wsierp_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => k5_wsierp_1(A,B) = k6_finseq_1(B) ) )).

fof(dt_k6_wsierp_1,axiom,
    ( ~ v1_xboole_0(k6_wsierp_1)
    & m1_subset_1(k6_wsierp_1,k1_zfmisc_1(k1_numbers)) )).

fof(redefinition_k6_wsierp_1,axiom,(
    k6_wsierp_1 = k4_numbers )).

fof(dt_k7_wsierp_1,axiom,(
    ! [A] :
      ( m1_finseq_1(A,k6_wsierp_1)
     => m2_subset_1(k7_wsierp_1(A),k1_numbers,k6_wsierp_1) ) )).

fof(redefinition_k7_wsierp_1,axiom,(
    ! [A] :
      ( m1_finseq_1(A,k6_wsierp_1)
     => k7_wsierp_1(A) = k15_rvsum_1(A) ) )).

fof(dt_k8_wsierp_1,axiom,(
    ! [A] :
      ( m1_finseq_1(A,k6_wsierp_1)
     => m2_subset_1(k8_wsierp_1(A),k1_numbers,k6_wsierp_1) ) )).

fof(redefinition_k8_wsierp_1,axiom,(
    ! [A] :
      ( m1_finseq_1(A,k6_wsierp_1)
     => k8_wsierp_1(A) = k16_rvsum_1(A) ) )).

fof(dt_k9_wsierp_1,axiom,(
    ! [A] :
      ( m1_finseq_1(A,k5_numbers)
     => m2_subset_1(k9_wsierp_1(A),k1_numbers,k5_numbers) ) )).

fof(redefinition_k9_wsierp_1,axiom,(
    ! [A] :
      ( m1_finseq_1(A,k5_numbers)
     => k9_wsierp_1(A) = k15_rvsum_1(A) ) )).

fof(dt_k10_wsierp_1,axiom,(
    ! [A] :
      ( m1_finseq_1(A,k5_numbers)
     => m2_subset_1(k10_wsierp_1(A),k1_numbers,k5_numbers) ) )).

fof(redefinition_k10_wsierp_1,axiom,(
    ! [A] :
      ( m1_finseq_1(A,k5_numbers)
     => k10_wsierp_1(A) = k16_rvsum_1(A) ) )).

fof(dt_k11_wsierp_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,k5_numbers)
        & m1_finseq_1(C,A) )
     => m2_finseq_1(k11_wsierp_1(A,B,C),A) ) )).

fof(redefinition_k11_wsierp_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,k5_numbers)
        & m1_finseq_1(C,A) )
     => k11_wsierp_1(A,B,C) = k2_finseq_3(B,C) ) )).

fof(dt_k12_wsierp_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k5_numbers)
        & m1_finseq_1(D,B) )
     => m1_trees_4(k12_wsierp_1(A,B,C,D),A,B) ) )).

fof(redefinition_k12_wsierp_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k5_numbers)
        & m1_finseq_1(D,B) )
     => k12_wsierp_1(A,B,C,D) = k2_finseq_3(C,D) ) )).
%------------------------------------------------------------------------------