SET007 Axioms: SET007+677.ax


%------------------------------------------------------------------------------
% File     : SET007+677 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Binomial Theorem for Algebraic Structures
% 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    : binom [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   57 (   2 unit)
%            Number of atoms       :  402 (  56 equality)
%            Maximal formula depth :   22 (   8 average)
%            Number of connectives :  400 (  55 ~  ;   0  |; 190  &)
%                                         (   5 <=>; 150 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   32 (   1 propositional; 0-3 arity)
%            Number of functors    :   43 (  13 constant; 0-6 arity)
%            Number of variables   :  155 (   0 singleton; 149 !;   6 ?)
%            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(rc1_binom,axiom,(
    ? [A] :
      ( l1_rlvect_1(A)
      & ~ v3_struct_0(A)
      & v2_algstr_1(A) ) )).

fof(rc2_binom,axiom,(
    ? [A] :
      ( l1_rlvect_1(A)
      & ~ v3_struct_0(A)
      & v3_algstr_1(A) ) )).

fof(rc3_binom,axiom,(
    ? [A] :
      ( l1_rlvect_1(A)
      & ~ v3_struct_0(A)
      & v1_binom(A) ) )).

fof(cc1_binom,axiom,(
    ! [A] :
      ( l1_rlvect_1(A)
     => ( ( ~ v3_struct_0(A)
          & v2_algstr_1(A)
          & v3_algstr_1(A) )
       => ( ~ v3_struct_0(A)
          & v1_binom(A) ) ) ) )).

fof(cc2_binom,axiom,(
    ! [A] :
      ( l1_rlvect_1(A)
     => ( ( ~ v3_struct_0(A)
          & v1_binom(A) )
       => ( ~ v3_struct_0(A)
          & v2_algstr_1(A)
          & v3_algstr_1(A) ) ) ) )).

fof(cc3_binom,axiom,(
    ! [A] :
      ( l1_rlvect_1(A)
     => ( ( ~ v3_struct_0(A)
          & v3_rlvect_1(A)
          & v3_algstr_1(A) )
       => ( ~ v3_struct_0(A)
          & v2_algstr_1(A) ) ) ) )).

fof(cc4_binom,axiom,(
    ! [A] :
      ( l1_rlvect_1(A)
     => ( ( ~ v3_struct_0(A)
          & v3_rlvect_1(A)
          & v2_algstr_1(A) )
       => ( ~ v3_struct_0(A)
          & v3_algstr_1(A) ) ) ) )).

fof(cc5_binom,axiom,(
    ! [A] :
      ( l1_rlvect_1(A)
     => ( ( ~ v3_struct_0(A)
          & v4_rlvect_1(A)
          & v5_rlvect_1(A)
          & v6_rlvect_1(A) )
       => ( ~ v3_struct_0(A)
          & v3_algstr_1(A) ) ) ) )).

fof(rc4_binom,axiom,(
    ? [A] :
      ( l3_vectsp_1(A)
      & ~ v3_struct_0(A)
      & v3_rlvect_1(A)
      & v4_rlvect_1(A)
      & v5_rlvect_1(A)
      & v2_group_1(A)
      & v4_group_1(A)
      & v7_group_1(A)
      & v4_vectsp_1(A)
      & v5_vectsp_1(A)
      & v6_vectsp_1(A)
      & v7_vectsp_1(A)
      & v8_vectsp_1(A)
      & v1_algstr_1(A)
      & v2_algstr_1(A)
      & v3_algstr_1(A)
      & v1_binom(A) ) )).

fof(d1_binom,axiom,(
    $true )).

fof(d2_binom,axiom,(
    $true )).

fof(d3_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ( v1_binom(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ( ( k2_rlvect_1(A,B,C) = k2_rlvect_1(A,B,D)
                       => C = D )
                      & ( k2_rlvect_1(A,C,B) = k2_rlvect_1(A,D,B)
                       => C = D ) ) ) ) ) ) ) )).

fof(t1_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & v5_vectsp_1(A)
        & v2_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k1_group_1(A,k1_rlvect_1(A),B) = k1_rlvect_1(A) ) ) )).

fof(t2_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_vectsp_1(A)
        & v1_algstr_1(A)
        & v3_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k1_group_1(A,B,k1_rlvect_1(A)) = k1_rlvect_1(A) ) ) )).

fof(t3_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_algstr_1(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k9_rlvect_1(A,k12_finseq_1(u1_struct_0(A),B)) = B ) ) )).

fof(t4_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_vectsp_1(A)
        & v1_algstr_1(A)
        & v3_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(A))
             => k9_rlvect_1(A,k6_polynom1(A,C,B)) = k1_group_1(A,B,k9_rlvect_1(A,C)) ) ) ) )).

fof(t5_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & v5_vectsp_1(A)
        & v2_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(A))
             => k9_rlvect_1(A,k7_polynom1(A,C,B)) = k1_group_1(A,k9_rlvect_1(A,C),B) ) ) ) )).

fof(t6_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v7_group_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(A))
             => k9_rlvect_1(A,k7_polynom1(A,C,B)) = k9_rlvect_1(A,k6_polynom1(A,C,B)) ) ) ) )).

fof(d4_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(A))
             => ( k4_relset_1(k5_numbers,u1_struct_0(A),B) = k4_relset_1(k5_numbers,u1_struct_0(A),C)
               => ! [D] :
                    ( m2_finseq_1(D,u1_struct_0(A))
                   => ( D = k1_binom(A,B,C)
                    <=> ( k4_relset_1(k5_numbers,u1_struct_0(A),D) = k4_relset_1(k5_numbers,u1_struct_0(A),B)
                        & ! [E] :
                            ( m2_subset_1(E,k1_numbers,k5_numbers)
                           => ( ( r1_xreal_0(np__1,E)
                                & r1_xreal_0(E,k3_finseq_1(D)) )
                             => k4_finseq_4(k5_numbers,u1_struct_0(A),D,E) = k2_rlvect_1(A,k4_finseq_4(k5_numbers,u1_struct_0(A),B,E),k4_finseq_4(k5_numbers,u1_struct_0(A),C,E)) ) ) ) ) ) ) ) ) ) )).

fof(t7_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(A))
             => ( k4_relset_1(k5_numbers,u1_struct_0(A),B) = k4_relset_1(k5_numbers,u1_struct_0(A),C)
               => k9_rlvect_1(A,k1_binom(A,B,C)) = k4_rlvect_1(A,k9_rlvect_1(A,B),k9_rlvect_1(A,C)) ) ) ) ) )).

fof(d5_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k2_binom(A,B,C) = k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),k5_group_1(A),B,C) ) ) ) )).

fof(t8_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( k2_binom(A,B,np__0) = k2_group_1(A)
            & k2_binom(A,B,np__1) = B ) ) ) )).

fof(t9_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k2_binom(A,B,k1_nat_1(C,np__1)) = k1_group_1(A,k2_binom(A,B,C),B) ) ) ) )).

fof(t10_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k2_binom(A,k10_group_1(A,B,C),D) = k10_group_1(A,k2_binom(A,B,D),k2_binom(A,C,D)) ) ) ) ) )).

fof(t11_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k2_binom(A,B,k1_nat_1(C,D)) = k1_group_1(A,k2_binom(A,B,C),k2_binom(A,B,D)) ) ) ) ) )).

fof(t12_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k2_binom(A,k2_binom(A,B,C),D) = k2_binom(A,B,k2_nat_1(C,D)) ) ) ) ) )).

fof(d6_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(k5_numbers,u1_struct_0(A)),u1_struct_0(A))
            & m2_relset_1(B,k2_zfmisc_1(k5_numbers,u1_struct_0(A)),u1_struct_0(A)) )
         => ( B = k3_binom(A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ( k2_binop_1(k5_numbers,u1_struct_0(A),u1_struct_0(A),B,np__0,C) = k1_rlvect_1(A)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => k2_binop_1(k5_numbers,u1_struct_0(A),u1_struct_0(A),B,k1_nat_1(D,np__1),C) = k2_rlvect_1(A,C,k2_binop_1(k5_numbers,u1_struct_0(A),u1_struct_0(A),B,D,C)) ) ) ) ) ) ) )).

fof(d7_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(u1_struct_0(A),k5_numbers),u1_struct_0(A))
            & m2_relset_1(B,k2_zfmisc_1(u1_struct_0(A),k5_numbers),u1_struct_0(A)) )
         => ( B = k4_binom(A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ( k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),B,C,np__0) = k1_rlvect_1(A)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),B,C,k1_nat_1(D,np__1)) = k2_rlvect_1(A,k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),B,C,D),C) ) ) ) ) ) ) )).

fof(d8_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k5_binom(A,B,C) = k2_binop_1(k5_numbers,u1_struct_0(A),u1_struct_0(A),k3_binom(A),C,B) ) ) ) )).

fof(d9_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k6_binom(A,B,C) = k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),k4_binom(A),B,C) ) ) ) )).

fof(t13_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( k5_binom(A,B,np__0) = k1_rlvect_1(A)
            & k6_binom(A,B,np__0) = k1_rlvect_1(A) ) ) ) )).

fof(t14_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k5_binom(A,B,np__1) = B ) ) )).

fof(t15_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_algstr_1(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k6_binom(A,B,np__1) = B ) ) )).

fof(t16_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v1_algstr_1(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k5_binom(A,B,k1_nat_1(C,D)) = k2_rlvect_1(A,k5_binom(A,B,C),k5_binom(A,B,D)) ) ) ) ) )).

fof(t17_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k6_binom(A,B,k1_nat_1(C,D)) = k2_rlvect_1(A,k6_binom(A,B,C),k6_binom(A,B,D)) ) ) ) ) )).

fof(t18_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v1_algstr_1(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k5_binom(A,B,C) = k6_binom(A,B,C) ) ) ) )).

fof(t19_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k5_binom(A,B,C) = k6_binom(A,B,C) ) ) ) )).

fof(t20_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v5_vectsp_1(A)
        & v1_algstr_1(A)
        & v2_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k1_group_1(A,k5_binom(A,B,D),C) = k5_binom(A,k1_group_1(A,B,C),D) ) ) ) ) )).

fof(t21_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v7_vectsp_1(A)
        & v1_algstr_1(A)
        & v3_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k1_group_1(A,C,k5_binom(A,B,D)) = k6_binom(A,k1_group_1(A,C,B),D) ) ) ) ) )).

fof(t22_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v7_vectsp_1(A)
        & v1_algstr_1(A)
        & v1_binom(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k1_group_1(A,k6_binom(A,B,D),C) = k1_group_1(A,B,k5_binom(A,C,D)) ) ) ) ) )).

fof(d10_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ! [E] :
                      ( m2_finseq_1(E,u1_struct_0(A))
                     => ( E = k8_binom(A,B,C,D)
                      <=> ( k3_finseq_1(E) = k1_nat_1(D,np__1)
                          & ! [F] :
                              ( m2_subset_1(F,k1_numbers,k5_numbers)
                             => ! [G] :
                                  ( m2_subset_1(G,k1_numbers,k5_numbers)
                                 => ! [H] :
                                      ( m2_subset_1(H,k1_numbers,k5_numbers)
                                     => ( ( r2_hidden(F,k4_relset_1(k5_numbers,u1_struct_0(A),E))
                                          & H = k6_xcmplx_0(F,np__1)
                                          & G = k6_xcmplx_0(D,H) )
                                       => k4_finseq_4(k5_numbers,u1_struct_0(A),E,F) = k1_group_1(A,k5_binom(A,k2_binom(A,B,G),k7_binom(H,D)),k2_binom(A,C,H)) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t23_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & v2_group_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k8_binom(A,B,C,np__0) = k12_finseq_1(u1_struct_0(A),k2_group_1(A)) ) ) ) )).

fof(t24_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & v2_group_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k1_funct_1(k8_binom(A,B,C,D),np__1) = k2_binom(A,B,D) ) ) ) ) )).

fof(t25_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & v2_group_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k1_funct_1(k8_binom(A,B,C,D),k1_nat_1(D,np__1)) = k2_binom(A,C,D) ) ) ) ) )).

fof(t26_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v7_vectsp_1(A)
        & v1_algstr_1(A)
        & v1_binom(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k2_binom(A,k4_rlvect_1(A,B,C),D) = k9_rlvect_1(A,k8_binom(A,B,C,D)) ) ) ) ) )).

fof(s1_binom,axiom,
    ( ( p1_s1_binom(f1_s1_binom)
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ( ( r1_xreal_0(f1_s1_binom,A)
              & p1_s1_binom(A) )
           => p1_s1_binom(k1_nat_1(A,np__1)) ) ) )
   => ! [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ( r1_xreal_0(f1_s1_binom,A)
         => p1_s1_binom(A) ) ) )).

fof(s2_binom,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & v1_funct_2(A,k2_zfmisc_1(k5_numbers,f1_s2_binom),f2_s2_binom)
      & m2_relset_1(A,k2_zfmisc_1(k5_numbers,f1_s2_binom),f2_s2_binom)
      & ! [B] :
          ( m1_subset_1(B,f1_s2_binom)
         => ( k2_binop_1(k5_numbers,f1_s2_binom,f2_s2_binom,A,np__0,B) = f3_s2_binom
            & ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k2_binop_1(k5_numbers,f1_s2_binom,f2_s2_binom,A,k1_nat_1(C,np__1),B) = k2_binop_1(f1_s2_binom,f2_s2_binom,f2_s2_binom,f4_s2_binom,B,k2_binop_1(k5_numbers,f1_s2_binom,f2_s2_binom,A,C,B)) ) ) ) ) )).

fof(s3_binom,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & v1_funct_2(A,k2_zfmisc_1(f1_s3_binom,k5_numbers),f2_s3_binom)
      & m2_relset_1(A,k2_zfmisc_1(f1_s3_binom,k5_numbers),f2_s3_binom)
      & ! [B] :
          ( m1_subset_1(B,f1_s3_binom)
         => ( k2_binop_1(f1_s3_binom,k5_numbers,f2_s3_binom,A,B,np__0) = f3_s3_binom
            & ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k2_binop_1(f1_s3_binom,k5_numbers,f2_s3_binom,A,B,k1_nat_1(C,np__1)) = k2_binop_1(f2_s3_binom,f1_s3_binom,f2_s3_binom,f4_s3_binom,k2_binop_1(f1_s3_binom,k5_numbers,f2_s3_binom,A,B,C),B) ) ) ) ) )).

fof(dt_k1_binom,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A)
        & m1_finseq_1(B,u1_struct_0(A))
        & m1_finseq_1(C,u1_struct_0(A)) )
     => m2_finseq_1(k1_binom(A,B,C),u1_struct_0(A)) ) )).

fof(dt_k2_binom,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l1_group_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,k5_numbers) )
     => m1_subset_1(k2_binom(A,B,C),u1_struct_0(A)) ) )).

fof(dt_k3_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ( v1_funct_1(k3_binom(A))
        & v1_funct_2(k3_binom(A),k2_zfmisc_1(k5_numbers,u1_struct_0(A)),u1_struct_0(A))
        & m2_relset_1(k3_binom(A),k2_zfmisc_1(k5_numbers,u1_struct_0(A)),u1_struct_0(A)) ) ) )).

fof(dt_k4_binom,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ( v1_funct_1(k4_binom(A))
        & v1_funct_2(k4_binom(A),k2_zfmisc_1(u1_struct_0(A),k5_numbers),u1_struct_0(A))
        & m2_relset_1(k4_binom(A),k2_zfmisc_1(u1_struct_0(A),k5_numbers),u1_struct_0(A)) ) ) )).

fof(dt_k5_binom,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,k5_numbers) )
     => m1_subset_1(k5_binom(A,B,C),u1_struct_0(A)) ) )).

fof(dt_k6_binom,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,k5_numbers) )
     => m1_subset_1(k6_binom(A,B,C),u1_struct_0(A)) ) )).

fof(dt_k7_binom,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => m2_subset_1(k7_binom(A,B),k1_numbers,k5_numbers) ) )).

fof(redefinition_k7_binom,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => k7_binom(A,B) = k7_newton(A,B) ) )).

fof(dt_k8_binom,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A))
        & m1_subset_1(D,k5_numbers) )
     => m2_finseq_1(k8_binom(A,B,C,D),u1_struct_0(A)) ) )).
%------------------------------------------------------------------------------