SET007 Axioms: SET007+132.ax


%------------------------------------------------------------------------------
% File     : SET007+132 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Factorial and Newton 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    : newton [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  139 (  22 unit)
%            Number of atoms       :  488 (  98 equality)
%            Maximal formula depth :   20 (   5 average)
%            Number of connectives :  395 (  46 ~  ;   2  |;  93  &)
%                                         (  18 <=>; 236 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   28 (   1 propositional; 0-3 arity)
%            Number of functors    :   51 (   8 constant; 0-3 arity)
%            Number of variables   :  229 (   0 singleton; 222 !;   7 ?)
%            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_newton,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v4_ordinal2(B) )
     => ( v1_xcmplx_0(k2_newton(A,B))
        & v1_xreal_0(k2_newton(A,B)) ) ) )).

fof(fc2_newton,axiom,(
    ! [A,B] :
      ( ( v4_ordinal2(A)
        & v4_ordinal2(B) )
     => ( v4_ordinal2(k2_newton(A,B))
        & v1_xcmplx_0(k2_newton(A,B))
        & v1_xreal_0(k2_newton(A,B))
        & ~ v3_xreal_0(k2_newton(A,B))
        & v1_int_1(k2_newton(A,B)) ) ) )).

fof(fc3_newton,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( v1_xcmplx_0(k5_newton(A))
        & v1_xreal_0(k5_newton(A)) ) ) )).

fof(fc4_newton,axiom,(
    ! [A,B] :
      ( ( v4_ordinal2(A)
        & v4_ordinal2(B) )
     => ( v1_xcmplx_0(k7_newton(A,B))
        & v1_xreal_0(k7_newton(A,B)) ) ) )).

fof(rc1_newton,axiom,(
    ? [A] :
      ( m1_subset_1(A,k5_numbers)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A)
      & v4_ordinal2(A)
      & v1_xcmplx_0(A)
      & v1_xreal_0(A)
      & ~ v3_xreal_0(A)
      & v1_int_1(A)
      & v1_int_2(A) ) )).

fof(fc5_newton,axiom,
    ( ~ v1_xboole_0(k12_newton)
    & ~ v1_finset_1(k12_newton)
    & v1_membered(k12_newton)
    & v2_membered(k12_newton)
    & v3_membered(k12_newton)
    & v4_membered(k12_newton)
    & v5_membered(k12_newton) )).

fof(t1_newton,axiom,(
    $true )).

fof(t2_newton,axiom,(
    $true )).

fof(t3_newton,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( ( k3_finseq_1(A) = k3_finseq_1(B)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( r2_hidden(C,k4_finseq_1(A))
                   => k1_funct_1(A,C) = k1_funct_1(B,C) ) ) )
           => A = B ) ) ) )).

fof(t4_newton,axiom,(
    $true )).

fof(t6_newton,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => k3_finseq_1(k9_rvsum_1(A,B)) = k3_finseq_1(B) ) ) )).

fof(t7_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( m2_finseq_1(C,k1_numbers)
             => ( r2_hidden(A,k4_finseq_1(C))
              <=> r2_hidden(A,k4_finseq_1(k9_rvsum_1(B,C))) ) ) ) ) )).

fof(d1_newton,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => k2_newton(A,B) = k16_rvsum_1(k1_newton(B,A)) ) ) )).

fof(t8_newton,axiom,(
    $true )).

fof(t9_newton,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k2_newton(A,np__0) = np__1 ) )).

fof(t10_newton,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k2_newton(A,np__1) = A ) )).

fof(t11_newton,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => k2_newton(B,k2_xcmplx_0(A,np__1)) = k3_xcmplx_0(k2_newton(B,A),B) ) ) )).

fof(t12_newton,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => k2_newton(k3_xcmplx_0(B,C),A) = k3_xcmplx_0(k2_newton(B,A),k2_newton(C,A)) ) ) ) )).

fof(t13_newton,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => k2_newton(C,k2_xcmplx_0(A,B)) = k3_xcmplx_0(k2_newton(C,A),k2_newton(C,B)) ) ) ) )).

fof(t14_newton,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => k2_newton(k2_newton(C,A),B) = k2_newton(C,k3_xcmplx_0(A,B)) ) ) ) )).

fof(t15_newton,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => k3_newton(np__1,A) = np__1 ) )).

fof(t16_newton,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( r1_xreal_0(np__1,A)
       => k3_newton(np__0,A) = np__0 ) ) )).

fof(d2_newton,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => k5_newton(A) = k16_rvsum_1(k4_newton(A)) ) )).

fof(t17_newton,axiom,(
    $true )).

fof(t18_newton,axiom,(
    k6_newton(np__0) = np__1 )).

fof(t19_newton,axiom,(
    k6_newton(np__1) = np__1 )).

fof(t20_newton,axiom,(
    k6_newton(np__2) = np__2 )).

fof(t21_newton,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => k6_newton(k2_xcmplx_0(A,np__1)) = k3_xcmplx_0(k6_newton(A),k2_xcmplx_0(A,np__1)) ) )).

fof(t22_newton,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => m2_subset_1(k6_newton(A),k1_numbers,k5_numbers) ) )).

fof(t23_newton,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ~ r1_xreal_0(k6_newton(A),np__0) ) )).

fof(t24_newton,axiom,(
    $true )).

fof(t25_newton,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => k3_xcmplx_0(k6_newton(A),k6_newton(B)) != np__0 ) ) )).

fof(d3_newton,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( ( r1_xreal_0(A,B)
               => ( C = k7_newton(A,B)
                <=> ! [D] :
                      ( v4_ordinal2(D)
                     => ( D = k6_xcmplx_0(B,A)
                       => C = k7_xcmplx_0(k6_newton(B),k3_xcmplx_0(k6_newton(A),k6_newton(D))) ) ) ) )
              & ( ~ r1_xreal_0(A,B)
               => ( C = k7_newton(A,B)
                <=> C = np__0 ) ) ) ) ) )).

fof(t26_newton,axiom,(
    $true )).

fof(t27_newton,axiom,(
    k8_newton(np__0,np__0) = np__1 )).

fof(t28_newton,axiom,(
    $true )).

fof(t29_newton,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => k8_newton(np__0,A) = np__1 ) )).

fof(t30_newton,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( r1_xreal_0(B,A)
           => ! [C] :
                ( v4_ordinal2(C)
               => ( C = k6_xcmplx_0(A,B)
                 => k8_newton(B,A) = k8_newton(C,A) ) ) ) ) ) )).

fof(t31_newton,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => k8_newton(A,A) = np__1 ) )).

fof(t32_newton,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => k8_newton(k2_xcmplx_0(B,np__1),k2_xcmplx_0(A,np__1)) = k2_xcmplx_0(k8_newton(k2_xcmplx_0(B,np__1),A),k8_newton(B,A)) ) ) )).

fof(t33_newton,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( r1_xreal_0(np__1,A)
       => k8_newton(np__1,A) = A ) ) )).

fof(t34_newton,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( ( r1_xreal_0(np__1,A)
              & B = k6_xcmplx_0(A,np__1) )
           => k8_newton(B,A) = A ) ) ) )).

fof(t35_newton,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => m2_subset_1(k8_newton(B,A),k1_numbers,k5_numbers) ) ) )).

fof(t36_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_finseq_1(C,k1_numbers)
             => ( ( k3_finseq_1(C) = B
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ( ( r2_hidden(D,k4_finseq_1(C))
                              & E = k6_xcmplx_0(k1_nat_1(A,D),np__1) )
                           => k1_funct_1(C,D) = k8_newton(A,E) ) ) ) )
               => ( B = np__0
                  | k15_rvsum_1(C) = k8_newton(k1_nat_1(A,np__1),k1_nat_1(A,B)) ) ) ) ) ) )).

fof(d4_newton,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => ! [D] :
                  ( m2_finseq_1(D,k1_numbers)
                 => ( D = k9_newton(A,B,C)
                  <=> ( k3_finseq_1(D) = k2_xcmplx_0(C,np__1)
                      & ! [E] :
                          ( v4_ordinal2(E)
                         => ! [F] :
                              ( v4_ordinal2(F)
                             => ! [G] :
                                  ( v4_ordinal2(G)
                                 => ( ( r2_hidden(E,k4_finseq_1(D))
                                      & G = k6_xcmplx_0(E,np__1)
                                      & F = k6_xcmplx_0(C,G) )
                                   => k1_funct_1(D,E) = k3_xcmplx_0(k3_xcmplx_0(k8_newton(G,C),k2_newton(A,F)),k2_newton(B,G)) ) ) ) ) ) ) ) ) ) ) )).

fof(t37_newton,axiom,(
    $true )).

fof(t38_newton,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => k9_newton(A,B,np__0) = k12_finseq_1(k5_numbers,np__1) ) ) )).

fof(t39_newton,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => k1_funct_1(k9_newton(B,C,A),np__1) = k2_newton(B,A) ) ) ) )).

fof(t40_newton,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => k1_funct_1(k9_newton(B,C,A),k2_xcmplx_0(A,np__1)) = k2_newton(C,A) ) ) ) )).

fof(t41_newton,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => k2_newton(k2_xcmplx_0(A,B),C) = k15_rvsum_1(k9_newton(A,B,C)) ) ) ) )).

fof(d5_newton,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => ( B = k10_newton(A)
          <=> ( k3_finseq_1(B) = k2_xcmplx_0(A,np__1)
              & ! [C] :
                  ( v4_ordinal2(C)
                 => ! [D] :
                      ( v4_ordinal2(D)
                     => ( ( r2_hidden(C,k4_finseq_1(B))
                          & D = k6_xcmplx_0(C,np__1) )
                       => k1_funct_1(B,C) = k8_newton(D,A) ) ) ) ) ) ) ) )).

fof(t42_newton,axiom,(
    $true )).

fof(t43_newton,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => k10_newton(A) = k9_newton(np__1,np__1,A) ) )).

fof(t44_newton,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => k3_newton(np__2,A) = k15_rvsum_1(k10_newton(A)) ) )).

fof(t45_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(np__1,A)
           => r1_xreal_0(B,k2_nat_1(B,A)) ) ) ) )).

fof(t46_newton,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(np__1,A)
                  & r1_xreal_0(k2_nat_1(C,A),B) )
               => r1_xreal_0(C,B) ) ) ) ) )).

fof(t47_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( A != np__0
       => r1_nat_1(A,k11_newton(A)) ) ) )).

fof(t48_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ~ ( A != np__0
          & r1_xreal_0(k7_xcmplx_0(k1_nat_1(A,np__1),A),np__1) ) ) )).

fof(t49_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ~ r1_xreal_0(np__1,k7_xcmplx_0(A,k1_nat_1(A,np__1))) ) )).

fof(t50_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => r1_xreal_0(A,k11_newton(A)) ) )).

fof(t51_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ~ ( A != np__1
              & r1_nat_1(A,B)
              & r1_nat_1(A,k1_nat_1(B,np__1)) ) ) ) )).

fof(t52_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ( r1_nat_1(B,A)
              & r1_nat_1(B,k1_nat_1(A,np__1)) )
          <=> B = np__1 ) ) ) )).

fof(t53_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( B != np__0
           => r1_nat_1(B,k11_newton(k1_nat_1(B,A))) ) ) ) )).

fof(t54_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(B,A)
           => ( B = np__0
              | r1_nat_1(B,k11_newton(A)) ) ) ) ) )).

fof(t55_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ~ ( B != np__1
              & B != np__0
              & r1_nat_1(B,k1_nat_1(k11_newton(A),np__1))
              & r1_xreal_0(B,A) ) ) ) )).

fof(t56_newton,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)
             => k5_nat_1(A,k5_nat_1(B,C)) = k5_nat_1(k5_nat_1(A,B),C) ) ) ) )).

fof(t57_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_nat_1(A,B)
          <=> k5_nat_1(A,B) = B ) ) ) )).

fof(t58_newton,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(C,B) )
              <=> r1_nat_1(k5_nat_1(A,C),B) ) ) ) ) )).

fof(t59_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k5_nat_1(A,np__1) = A ) )).

fof(t60_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => r1_nat_1(k5_nat_1(A,B),k2_nat_1(A,B)) ) ) )).

fof(t61_newton,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,k6_nat_1(B,C)) = k6_nat_1(k6_nat_1(A,B),C) ) ) ) )).

fof(t62_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_nat_1(A,B)
           => k6_nat_1(A,B) = A ) ) ) )).

fof(t63_newton,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(A,C) )
              <=> r1_nat_1(A,k6_nat_1(B,C)) ) ) ) ) )).

fof(t64_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k6_nat_1(A,np__1) = np__1 ) )).

fof(t65_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k6_nat_1(A,np__0) = A ) )).

fof(t66_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k5_nat_1(k6_nat_1(A,B),B) = B ) ) )).

fof(t67_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k6_nat_1(A,k5_nat_1(A,B)) = A ) ) )).

fof(t68_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k6_nat_1(A,k5_nat_1(A,B)) = k5_nat_1(k6_nat_1(B,A),A) ) ) )).

fof(t69_newton,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(k6_nat_1(A,C),k6_nat_1(B,C)) ) ) ) ) )).

fof(t70_newton,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(k6_nat_1(C,A),k6_nat_1(C,B)) ) ) ) ) )).

fof(t71_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ~ ( ~ r1_xreal_0(A,np__0)
              & r1_xreal_0(k6_nat_1(A,B),np__0) ) ) ) )).

fof(t72_newton,axiom,(
    $true )).

fof(t73_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ~ ( ~ r1_xreal_0(A,np__0)
              & ~ r1_xreal_0(B,np__0)
              & r1_xreal_0(k5_nat_1(A,B),np__0) ) ) ) )).

fof(t74_newton,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(k5_nat_1(k6_nat_1(A,B),k6_nat_1(A,C)),k6_nat_1(A,k5_nat_1(B,C))) ) ) ) )).

fof(t75_newton,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(k5_nat_1(A,k6_nat_1(C,B)),k6_nat_1(k5_nat_1(A,C),B)) ) ) ) ) )).

fof(t76_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => r1_nat_1(k6_nat_1(A,B),k5_nat_1(A,B)) ) ) )).

fof(t77_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ~ r1_xreal_0(B,np__0)
           => k4_nat_1(A,B) = k6_xcmplx_0(A,k2_nat_1(B,k3_nat_1(A,B))) ) ) ) )).

fof(t78_newton,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ( r1_xreal_0(np__0,A)
           => r1_xreal_0(np__0,k6_int_1(B,A)) ) ) ) )).

fof(t79_newton,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ~ ( ~ r1_xreal_0(A,np__0)
              & r1_xreal_0(A,k6_int_1(B,A)) ) ) ) )).

fof(t80_newton,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ( A != np__0
           => B = k2_xcmplx_0(k3_xcmplx_0(k5_int_1(B,A),A),k6_int_1(B,A)) ) ) ) )).

fof(t81_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ~ ( ~ ( r1_xreal_0(A,np__0)
                  & r1_xreal_0(B,np__0) )
              & ! [C] :
                  ( v1_int_1(C)
                 => ! [D] :
                      ( v1_int_1(D)
                     => k2_xcmplx_0(k3_xcmplx_0(C,A),k3_xcmplx_0(D,B)) != k6_nat_1(A,B) ) ) ) ) ) )).

fof(d6_newton,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k5_numbers))
     => ( A = k12_newton
      <=> ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( r2_hidden(B,A)
            <=> v1_int_2(B) ) ) ) ) )).

fof(d7_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k5_numbers))
         => ( B = k13_newton(A)
          <=> ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( r2_hidden(C,B)
                <=> ( ~ r1_xreal_0(A,C)
                    & v1_int_2(C) ) ) ) ) ) ) )).

fof(t82_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => r1_tarski(k13_newton(A),k12_newton) ) )).

fof(t83_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ~ r1_xreal_0(B,A)
           => r1_tarski(k13_newton(A),k13_newton(B)) ) ) ) )).

fof(t84_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => r1_tarski(k13_newton(A),k2_finseq_1(A)) ) )).

fof(t85_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => v1_finset_1(k13_newton(A)) ) )).

fof(t86_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ? [B] :
          ( v1_int_2(B)
          & m2_subset_1(B,k1_numbers,k5_numbers)
          & v1_int_2(B)
          & ~ r1_xreal_0(B,A) ) ) )).

fof(t87_newton,axiom,(
    k12_newton != k1_xboole_0 )).

fof(t97_newton,axiom,(
    ~ v1_finset_1(k12_newton) )).

fof(t98_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( v1_int_2(A)
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ~ ( r1_nat_1(A,k2_nat_1(B,C))
                    & ~ r1_nat_1(A,B)
                    & ~ r1_nat_1(A,C) ) ) ) ) ) )).

fof(t99_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => m2_subset_1(k3_newton(B,A),k1_numbers,k5_numbers) ) ) )).

fof(t100_newton,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( k2_newton(A,np__2) = k3_xcmplx_0(A,A)
        & k5_square_1(A) = k2_newton(A,np__2) ) ) )).

fof(t101_newton,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( k5_int_1(A,B) = k3_nat_1(A,B)
            & k6_int_1(A,B) = k4_nat_1(A,B) ) ) ) )).

fof(dt_k1_newton,axiom,(
    ! [A,B] :
      ( ( v4_ordinal2(A)
        & v1_xreal_0(B) )
     => m2_finseq_1(k1_newton(A,B),k1_numbers) ) )).

fof(redefinition_k1_newton,axiom,(
    ! [A,B] :
      ( ( v4_ordinal2(A)
        & v1_xreal_0(B) )
     => k1_newton(A,B) = k2_finseq_2(A,B) ) )).

fof(dt_k2_newton,axiom,(
    $true )).

fof(dt_k3_newton,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & v4_ordinal2(B) )
     => m1_subset_1(k3_newton(A,B),k1_numbers) ) )).

fof(redefinition_k3_newton,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & v4_ordinal2(B) )
     => k3_newton(A,B) = k2_newton(A,B) ) )).

fof(dt_k4_newton,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => m2_finseq_1(k4_newton(A),k1_numbers) ) )).

fof(redefinition_k4_newton,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => k4_newton(A) = k1_finseq_2(A) ) )).

fof(dt_k5_newton,axiom,(
    $true )).

fof(dt_k6_newton,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => m1_subset_1(k6_newton(A),k1_numbers) ) )).

fof(redefinition_k6_newton,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => k6_newton(A) = k5_newton(A) ) )).

fof(dt_k7_newton,axiom,(
    $true )).

fof(dt_k8_newton,axiom,(
    ! [A,B] :
      ( ( v4_ordinal2(A)
        & v4_ordinal2(B) )
     => m1_subset_1(k8_newton(A,B),k1_numbers) ) )).

fof(redefinition_k8_newton,axiom,(
    ! [A,B] :
      ( ( v4_ordinal2(A)
        & v4_ordinal2(B) )
     => k8_newton(A,B) = k7_newton(A,B) ) )).

fof(dt_k9_newton,axiom,(
    ! [A,B,C] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B)
        & v4_ordinal2(C) )
     => m2_finseq_1(k9_newton(A,B,C),k1_numbers) ) )).

fof(dt_k10_newton,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => m2_finseq_1(k10_newton(A),k1_numbers) ) )).

fof(dt_k11_newton,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => m2_subset_1(k11_newton(A),k1_numbers,k5_numbers) ) )).

fof(redefinition_k11_newton,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => k11_newton(A) = k5_newton(A) ) )).

fof(dt_k12_newton,axiom,(
    m1_subset_1(k12_newton,k1_zfmisc_1(k5_numbers)) )).

fof(dt_k13_newton,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => m1_subset_1(k13_newton(A),k1_zfmisc_1(k5_numbers)) ) )).

fof(dt_k14_newton,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ( v1_int_2(k14_newton(A))
        & m2_subset_1(k14_newton(A),k1_numbers,k5_numbers) ) ) )).

fof(t5_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( r1_xreal_0(np__1,A)
       => k2_finseq_1(A) = k2_xboole_0(k2_xboole_0(k1_tarski(np__1),a_1_0_newton(A)),k1_tarski(A)) ) ) )).

fof(t88_newton,axiom,(
    a_0_0_newton = k1_xboole_0 )).

fof(t89_newton,axiom,(
    ! [A] :
      ( ( v1_int_2(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => r1_tarski(a_1_1_newton(A),k5_numbers) ) )).

fof(t90_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => r1_tarski(a_1_2_newton(A),k2_finseq_1(A)) ) )).

fof(t91_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => v1_finset_1(a_1_2_newton(A)) ) )).

fof(t92_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ~ r2_hidden(A,a_1_2_newton(A)) ) )).

fof(t93_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => v1_finset_1(k2_xboole_0(a_1_2_newton(A),k1_tarski(A))) ) )).

fof(t94_newton,axiom,(
    ! [A] :
      ( ( v1_int_2(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( v1_int_2(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ( ~ r1_xreal_0(B,A)
           => r1_tarski(k2_xboole_0(a_1_1_newton(A),k1_tarski(A)),a_1_1_newton(B)) ) ) ) )).

fof(t95_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ~ ( ~ r1_xreal_0(B,A)
              & r2_hidden(B,a_1_2_newton(A)) ) ) ) )).

fof(d8_newton,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_int_2(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ( B = k14_newton(A)
          <=> ? [C] :
                ( v1_finset_1(C)
                & C = a_1_1_newton(B)
                & A = k4_card_1(C) ) ) ) ) )).

fof(t96_newton,axiom,(
    ! [A] :
      ( ( v1_int_2(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => k13_newton(A) = a_1_1_newton(A) ) )).

fof(fraenkel_a_1_0_newton,axiom,(
    ! [A,B] :
      ( m2_subset_1(B,k1_numbers,k5_numbers)
     => ( r2_hidden(A,a_1_0_newton(B))
      <=> ? [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
            & A = C
            & ~ r1_xreal_0(C,np__1)
            & ~ r1_xreal_0(B,C) ) ) ) )).

fof(fraenkel_a_0_0_newton,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_0_newton)
    <=> ? [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
          & A = B
          & ~ r1_xreal_0(np__2,B)
          & v1_int_2(B) ) ) )).

fof(fraenkel_a_1_1_newton,axiom,(
    ! [A,B] :
      ( ( v1_int_2(B)
        & m2_subset_1(B,k1_numbers,k5_numbers) )
     => ( r2_hidden(A,a_1_1_newton(B))
      <=> ? [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
            & A = C
            & ~ r1_xreal_0(B,C)
            & v1_int_2(C) ) ) ) )).

fof(fraenkel_a_1_2_newton,axiom,(
    ! [A,B] :
      ( m2_subset_1(B,k1_numbers,k5_numbers)
     => ( r2_hidden(A,a_1_2_newton(B))
      <=> ? [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
            & A = C
            & ~ r1_xreal_0(B,C)
            & v1_int_2(C) ) ) ) )).
%------------------------------------------------------------------------------