SET007 Axioms: SET007+685.ax


%------------------------------------------------------------------------------
% File     : SET007+685 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Algebra of Polynomials
% 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    : polyalg1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   60 (   1 unit)
%            Number of atoms       :  601 (  57 equality)
%            Maximal formula depth :   23 (  10 average)
%            Number of connectives :  632 (  91 ~  ;   0  |; 378  &)
%                                         (   8 <=>; 155 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   40 (   1 propositional; 0-3 arity)
%            Number of functors    :   38 (   2 constant; 0-7 arity)
%            Number of variables   :  183 (   0 singleton; 171 !;  12 ?)
%            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_polyalg1,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ? [B] :
          ( l1_polyalg1(B,A)
          & v1_polyalg1(B,A) ) ) )).

fof(rc2_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ? [B] :
          ( l1_polyalg1(B,A)
          & ~ v3_struct_0(B)
          & v1_polyalg1(B,A) ) ) )).

fof(rc3_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ? [B] :
          ( l1_polyalg1(B,A)
          & ~ v3_struct_0(B)
          & v2_group_1(B)
          & v4_vectsp_1(B)
          & v5_vectsp_1(B)
          & v6_vectsp_1(B)
          & v7_vectsp_1(B)
          & v8_vectsp_1(B)
          & v12_vectsp_1(B,A)
          & v2_polyalg1(B,A) ) ) )).

fof(fc1_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & l3_vectsp_1(A) )
     => ( ~ v3_struct_0(k1_polyalg1(A))
        & v3_rlvect_1(k1_polyalg1(A))
        & v1_polyalg1(k1_polyalg1(A),A) ) ) )).

fof(fc2_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & l3_vectsp_1(A) )
     => ( ~ v3_struct_0(k1_polyalg1(A))
        & v4_rlvect_1(k1_polyalg1(A))
        & v1_polyalg1(k1_polyalg1(A),A) ) ) )).

fof(fc3_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & l3_vectsp_1(A) )
     => ( ~ v3_struct_0(k1_polyalg1(A))
        & v5_rlvect_1(k1_polyalg1(A))
        & v1_polyalg1(k1_polyalg1(A),A) ) ) )).

fof(fc4_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & l3_vectsp_1(A) )
     => ( ~ v3_struct_0(k1_polyalg1(A))
        & v4_rlvect_1(k1_polyalg1(A))
        & v5_rlvect_1(k1_polyalg1(A))
        & v6_rlvect_1(k1_polyalg1(A))
        & v3_algstr_1(k1_polyalg1(A))
        & v1_polyalg1(k1_polyalg1(A),A) ) ) )).

fof(fc5_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v7_group_1(A)
        & l3_vectsp_1(A) )
     => ( ~ v3_struct_0(k1_polyalg1(A))
        & v3_rlvect_1(k1_polyalg1(A))
        & v4_rlvect_1(k1_polyalg1(A))
        & v5_rlvect_1(k1_polyalg1(A))
        & v7_group_1(k1_polyalg1(A))
        & v1_polyalg1(k1_polyalg1(A),A) ) ) )).

fof(fc6_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ( ~ v3_struct_0(k1_polyalg1(A))
        & v3_rlvect_1(k1_polyalg1(A))
        & v4_rlvect_1(k1_polyalg1(A))
        & v5_rlvect_1(k1_polyalg1(A))
        & v6_rlvect_1(k1_polyalg1(A))
        & v4_group_1(k1_polyalg1(A))
        & v2_algstr_1(k1_polyalg1(A))
        & v3_algstr_1(k1_polyalg1(A))
        & v1_polyalg1(k1_polyalg1(A),A) ) ) )).

fof(rc4_polyalg1,axiom,(
    ? [A] :
      ( l3_vectsp_1(A)
      & ~ v3_struct_0(A)
      & v4_rlvect_1(A)
      & v5_rlvect_1(A)
      & v6_rlvect_1(A)
      & v2_group_1(A)
      & v4_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)
      & v3_algstr_1(A) ) )).

fof(fc7_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v2_group_1(A)
        & v7_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ( ~ v3_struct_0(k1_polyalg1(A))
        & v4_rlvect_1(k1_polyalg1(A))
        & v5_rlvect_1(k1_polyalg1(A))
        & v6_rlvect_1(k1_polyalg1(A))
        & v2_group_1(k1_polyalg1(A))
        & v6_vectsp_1(k1_polyalg1(A))
        & v8_vectsp_1(k1_polyalg1(A))
        & v3_algstr_1(k1_polyalg1(A))
        & v1_polyalg1(k1_polyalg1(A),A) ) ) )).

fof(fc8_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ( ~ v3_struct_0(k1_polyalg1(A))
        & v3_rlvect_1(k1_polyalg1(A))
        & v4_rlvect_1(k1_polyalg1(A))
        & v5_rlvect_1(k1_polyalg1(A))
        & v6_rlvect_1(k1_polyalg1(A))
        & v4_vectsp_1(k1_polyalg1(A))
        & v2_algstr_1(k1_polyalg1(A))
        & v3_algstr_1(k1_polyalg1(A))
        & v1_polyalg1(k1_polyalg1(A),A) ) ) )).

fof(fc9_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ( ~ v3_struct_0(k1_polyalg1(A))
        & v3_rlvect_1(k1_polyalg1(A))
        & v4_rlvect_1(k1_polyalg1(A))
        & v5_rlvect_1(k1_polyalg1(A))
        & v6_rlvect_1(k1_polyalg1(A))
        & v5_vectsp_1(k1_polyalg1(A))
        & v2_algstr_1(k1_polyalg1(A))
        & v3_algstr_1(k1_polyalg1(A))
        & v1_polyalg1(k1_polyalg1(A),A) ) ) )).

fof(fc10_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ( ~ v3_struct_0(k1_polyalg1(A))
        & v3_rlvect_1(k1_polyalg1(A))
        & v4_rlvect_1(k1_polyalg1(A))
        & v5_rlvect_1(k1_polyalg1(A))
        & v6_rlvect_1(k1_polyalg1(A))
        & v4_vectsp_1(k1_polyalg1(A))
        & v5_vectsp_1(k1_polyalg1(A))
        & v7_vectsp_1(k1_polyalg1(A))
        & v12_vectsp_1(k1_polyalg1(A),A)
        & v2_algstr_1(k1_polyalg1(A))
        & v3_algstr_1(k1_polyalg1(A))
        & v1_polyalg1(k1_polyalg1(A),A) ) ) )).

fof(fc11_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v7_vectsp_1(A)
        & v1_algstr_1(A)
        & l3_vectsp_1(A) )
     => ( ~ v3_struct_0(k1_polyalg1(A))
        & v3_rlvect_1(k1_polyalg1(A))
        & v4_rlvect_1(k1_polyalg1(A))
        & v5_rlvect_1(k1_polyalg1(A))
        & v6_rlvect_1(k1_polyalg1(A))
        & v4_vectsp_1(k1_polyalg1(A))
        & v5_vectsp_1(k1_polyalg1(A))
        & v7_vectsp_1(k1_polyalg1(A))
        & v2_algstr_1(k1_polyalg1(A))
        & v3_algstr_1(k1_polyalg1(A))
        & v1_polyalg1(k1_polyalg1(A),A)
        & v2_polyalg1(k1_polyalg1(A),A) ) ) )).

fof(rc5_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ? [B] :
          ( l1_polyalg1(B,A)
          & ~ v3_struct_0(B)
          & v1_polyalg1(B,A) ) ) )).

fof(rc6_polyalg1,axiom,(
    ! [A,B] :
      ( ( l1_struct_0(A)
        & l1_polyalg1(B,A) )
     => ? [C] :
          ( m1_polyalg1(C,A,B)
          & v1_polyalg1(C,A) ) ) )).

fof(rc7_polyalg1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A)
        & ~ v3_struct_0(B)
        & l1_polyalg1(B,A) )
     => ? [C] :
          ( m1_polyalg1(C,A,B)
          & ~ v3_struct_0(C)
          & v1_polyalg1(C,A) ) ) )).

fof(fc12_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ( ~ v3_struct_0(k16_polynom3(A))
        & v4_rlvect_1(k16_polynom3(A))
        & v5_rlvect_1(k16_polynom3(A))
        & v6_rlvect_1(k16_polynom3(A))
        & v3_vectsp_1(k16_polynom3(A))
        & v3_algstr_1(k16_polynom3(A))
        & v6_algstr_1(k16_polynom3(A)) ) ) )).

fof(d1_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_polyalg1(B,A) )
         => ( v2_polyalg1(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(B))
                   => ! [E] :
                        ( m1_subset_1(E,u1_struct_0(B))
                       => k6_vectsp_1(A,B,C,k1_group_1(B,D,E)) = k1_group_1(B,k6_vectsp_1(A,B,C,D),E) ) ) ) ) ) ) )).

fof(t1_polyalg1,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(C)
        & v1_funct_2(C,k2_zfmisc_1(A,B),A)
        & m2_relset_1(C,k2_zfmisc_1(A,B),A) )
     => k1_relat_1(C) = k2_zfmisc_1(A,B) ) )).

fof(t2_polyalg1,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(C)
        & v1_funct_2(C,k2_zfmisc_1(A,B),B)
        & m2_relset_1(C,k2_zfmisc_1(A,B),B) )
     => k1_relat_1(C) = k2_zfmisc_1(A,B) ) )).

fof(d2_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_polyalg1(B,A)
            & l1_polyalg1(B,A) )
         => ( B = k1_polyalg1(A)
          <=> ( ! [C] :
                  ( r2_hidden(C,u1_struct_0(B))
                <=> ( v1_funct_1(C)
                    & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                    & m2_relset_1(C,k5_numbers,u1_struct_0(A)) ) )
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(B))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(B))
                     => ! [E] :
                          ( ( v1_funct_1(E)
                            & v1_funct_2(E,k5_numbers,u1_struct_0(A))
                            & m2_relset_1(E,k5_numbers,u1_struct_0(A)) )
                         => ! [F] :
                              ( ( v1_funct_1(F)
                                & v1_funct_2(F,k5_numbers,u1_struct_0(A))
                                & m2_relset_1(F,k5_numbers,u1_struct_0(A)) )
                             => ( ( C = E
                                  & D = F )
                               => k2_rlvect_1(B,C,D) = k8_polynom3(A,E,F) ) ) ) ) )
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(B))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(B))
                     => ! [E] :
                          ( ( v1_funct_1(E)
                            & v1_funct_2(E,k5_numbers,u1_struct_0(A))
                            & m2_relset_1(E,k5_numbers,u1_struct_0(A)) )
                         => ! [F] :
                              ( ( v1_funct_1(F)
                                & v1_funct_2(F,k5_numbers,u1_struct_0(A))
                                & m2_relset_1(F,k5_numbers,u1_struct_0(A)) )
                             => ( ( C = E
                                  & D = F )
                               => k1_group_1(B,C,D) = k14_polynom3(A,E,F) ) ) ) ) )
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(B))
                     => ! [E] :
                          ( ( v1_funct_1(E)
                            & v1_funct_2(E,k5_numbers,u1_struct_0(A))
                            & m2_relset_1(E,k5_numbers,u1_struct_0(A)) )
                         => ( D = E
                           => k6_vectsp_1(A,B,C,D) = k3_polynom5(A,E,C) ) ) ) )
              & k1_rlvect_1(B) = k12_polynom3(A)
              & k2_vectsp_1(B) = k13_polynom3(A) ) ) ) ) )).

fof(t3_polyalg1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_finseq_1(B,A) )
         => k1_rfinseq(A,B,np__1) = k1_polynom3(A,np__1,B) ) ) )).

fof(t4_polyalg1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_finseq_1(B,A) )
         => B = k7_finseq_1(k9_finseq_1(k1_funct_1(B,np__1)),k1_polynom3(A,np__1,B)) ) ) )).

fof(t5_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v5_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => k14_polynom3(A,k13_polynom3(A),B) = B ) ) )).

fof(t6_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(A))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                 => k3_polynom5(A,k9_polynom3(A,C,D),B) = k9_polynom3(A,k3_polynom5(A,C,B),k3_polynom5(A,D,B)) ) ) ) ) )).

fof(t7_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_vectsp_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] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(A))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                 => k3_polynom5(A,D,k4_rlvect_1(A,B,C)) = k9_polynom3(A,k3_polynom5(A,D,B),k3_polynom5(A,D,C)) ) ) ) ) )).

fof(t8_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_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] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(A))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                 => k3_polynom5(A,D,k1_group_1(A,B,C)) = k3_polynom5(A,k3_polynom5(A,D,C),B) ) ) ) ) )).

fof(t9_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => k3_polynom5(A,B,k2_group_1(A)) = B ) ) )).

fof(t10_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v7_vectsp_1(A)
        & v1_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(A))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                 => k3_polynom5(A,k14_polynom3(A,C,D),B) = k14_polynom3(A,k3_polynom5(A,C,B),D) ) ) ) ) )).

fof(d3_polyalg1,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( l1_polyalg1(B,A)
         => ! [C] :
              ( l1_polyalg1(C,A)
             => ( m1_polyalg1(C,A,B)
              <=> ( r1_tarski(u1_struct_0(C),u1_struct_0(B))
                  & k2_vectsp_1(C) = k2_vectsp_1(B)
                  & k1_rlvect_1(C) = k1_rlvect_1(B)
                  & u1_rlvect_1(C) = k1_realset1(u1_rlvect_1(B),u1_struct_0(C))
                  & u1_group_1(C) = k1_realset1(u1_group_1(B),u1_struct_0(C))
                  & u2_vectsp_1(A,C) = k7_relat_1(u2_vectsp_1(A,B),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(C))) ) ) ) ) ) )).

fof(t11_polyalg1,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( l1_polyalg1(B,A)
         => m1_polyalg1(B,A,B) ) ) )).

fof(t12_polyalg1,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( l1_polyalg1(B,A)
         => ! [C] :
              ( l1_polyalg1(C,A)
             => ! [D] :
                  ( l1_polyalg1(D,A)
                 => ( ( m1_polyalg1(B,A,C)
                      & m1_polyalg1(C,A,D) )
                   => m1_polyalg1(B,A,D) ) ) ) ) ) )).

fof(t13_polyalg1,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( l1_polyalg1(B,A)
         => ! [C] :
              ( l1_polyalg1(C,A)
             => ( ( m1_polyalg1(B,A,C)
                  & m1_polyalg1(C,A,B) )
               => g1_polyalg1(A,u1_struct_0(B),u1_rlvect_1(B),u1_group_1(B),u2_struct_0(B),u1_vectsp_1(B),u2_vectsp_1(A,B)) = g1_polyalg1(A,u1_struct_0(C),u1_rlvect_1(C),u1_group_1(C),u2_struct_0(C),u1_vectsp_1(C),u2_vectsp_1(A,C)) ) ) ) ) )).

fof(t14_polyalg1,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( l1_polyalg1(B,A)
         => ! [C] :
              ( l1_polyalg1(C,A)
             => ( g1_polyalg1(A,u1_struct_0(B),u1_rlvect_1(B),u1_group_1(B),u2_struct_0(B),u1_vectsp_1(B),u2_vectsp_1(A,B)) = g1_polyalg1(A,u1_struct_0(C),u1_rlvect_1(C),u1_group_1(C),u2_struct_0(C),u1_vectsp_1(C),u2_vectsp_1(A,C))
               => ( m1_polyalg1(B,A,C)
                  & m1_polyalg1(C,A,B) ) ) ) ) ) )).

fof(d4_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_polyalg1(B,A) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
             => ( v3_polyalg1(C,A,B)
              <=> ( v1_vectsp_4(C,A,B)
                  & ! [D] :
                      ( m1_subset_1(D,u1_struct_0(B))
                     => ! [E] :
                          ( m1_subset_1(E,u1_struct_0(B))
                         => ( ( r2_hidden(D,C)
                              & r2_hidden(E,C) )
                           => r2_hidden(k1_group_1(B,D,E),C) ) ) )
                  & r2_hidden(k2_vectsp_1(B),C)
                  & r2_hidden(k1_rlvect_1(B),C) ) ) ) ) ) )).

fof(t15_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_polyalg1(B,A) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & m1_polyalg1(C,A,B) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(B))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(C))
                         => ! [G] :
                              ( m1_subset_1(G,u1_struct_0(C))
                             => ( ( D = F
                                  & E = G )
                               => k2_rlvect_1(B,D,E) = k2_rlvect_1(C,F,G) ) ) ) ) ) ) ) ) )).

fof(t16_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_polyalg1(B,A) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & m1_polyalg1(C,A,B) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(B))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(C))
                         => ! [G] :
                              ( m1_subset_1(G,u1_struct_0(C))
                             => ( ( D = F
                                  & E = G )
                               => k1_group_1(B,D,E) = k1_group_1(C,F,G) ) ) ) ) ) ) ) ) )).

fof(t17_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_polyalg1(B,A) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & m1_polyalg1(C,A,B) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(B))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(C))
                         => ( E = F
                           => k6_vectsp_1(A,B,D,E) = k6_vectsp_1(A,C,D,F) ) ) ) ) ) ) ) )).

fof(t18_polyalg1,axiom,(
    $true )).

fof(t19_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_polyalg1(B,A) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & m1_polyalg1(C,A,B) )
             => ? [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
                  & u1_struct_0(C) = D
                  & v3_polyalg1(D,A,B) ) ) ) ) )).

fof(t20_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_polyalg1(B,A) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
             => ~ ( v3_polyalg1(C,A,B)
                  & ! [D] :
                      ( ( v1_polyalg1(D,A)
                        & m1_polyalg1(D,A,B) )
                     => u1_struct_0(D) != C ) ) ) ) ) )).

fof(t21_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_polyalg1(B,A) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B))))
                 => ( ! [E] :
                        ( r2_hidden(E,D)
                      <=> ( r1_tarski(E,u1_struct_0(B))
                          & ? [F] :
                              ( m1_polyalg1(F,A,B)
                              & E = u1_struct_0(F)
                              & r1_tarski(C,E) ) ) )
                   => v3_polyalg1(k6_setfam_1(u1_struct_0(B),D),A,B) ) ) ) ) ) )).

fof(d5_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_polyalg1(B,A) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
             => ! [D] :
                  ( ( ~ v3_struct_0(D)
                    & v1_polyalg1(D,A)
                    & m1_polyalg1(D,A,B) )
                 => ( D = k2_polyalg1(A,B,C)
                  <=> ( r1_tarski(C,u1_struct_0(D))
                      & ! [E] :
                          ( m1_polyalg1(E,A,B)
                         => ( r1_tarski(C,u1_struct_0(E))
                           => r1_tarski(u1_struct_0(D),u1_struct_0(E)) ) ) ) ) ) ) ) ) )).

fof(t22_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_polyalg1(B,A) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
             => ( v3_polyalg1(C,A,B)
               => u1_struct_0(k2_polyalg1(A,B,C)) = C ) ) ) ) )).

fof(d6_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_polyalg1(B,A)
            & l1_polyalg1(B,A) )
         => ( B = k3_polyalg1(A)
          <=> ? [C] :
                ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k1_polyalg1(A))))
                & C = u1_struct_0(k16_polynom3(A))
                & B = k2_polyalg1(A,k1_polyalg1(A),C) ) ) ) ) )).

fof(t23_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k1_polyalg1(A)))) )
         => ( B = u1_struct_0(k16_polynom3(A))
           => v3_polyalg1(B,A,k1_polyalg1(A)) ) ) ) )).

fof(t24_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_vectsp_1(A)
        & l3_vectsp_1(A) )
     => g3_vectsp_1(u1_struct_0(k3_polyalg1(A)),u1_rlvect_1(k3_polyalg1(A)),u1_group_1(k3_polyalg1(A)),u1_vectsp_1(k3_polyalg1(A)),u2_struct_0(k3_polyalg1(A))) = k16_polynom3(A) ) )).

fof(t25_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v2_group_1(A)
        & v7_vectsp_1(A)
        & l3_vectsp_1(A) )
     => k2_group_1(k1_polyalg1(A)) = k13_polynom3(A) ) )).

fof(dt_m1_polyalg1,axiom,(
    ! [A,B] :
      ( ( l1_struct_0(A)
        & l1_polyalg1(B,A) )
     => ! [C] :
          ( m1_polyalg1(C,A,B)
         => l1_polyalg1(C,A) ) ) )).

fof(existence_m1_polyalg1,axiom,(
    ! [A,B] :
      ( ( l1_struct_0(A)
        & l1_polyalg1(B,A) )
     => ? [C] : m1_polyalg1(C,A,B) ) )).

fof(dt_l1_polyalg1,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( l1_polyalg1(B,A)
         => ( l3_vectsp_1(B)
            & l4_vectsp_1(B,A) ) ) ) )).

fof(existence_l1_polyalg1,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ? [B] : l1_polyalg1(B,A) ) )).

fof(abstractness_v1_polyalg1,axiom,(
    ! [A,B] :
      ( ( l1_struct_0(A)
        & l1_polyalg1(B,A) )
     => ( v1_polyalg1(B,A)
       => B = g1_polyalg1(A,u1_struct_0(B),u1_rlvect_1(B),u1_group_1(B),u2_struct_0(B),u1_vectsp_1(B),u2_vectsp_1(A,B)) ) ) )).

fof(dt_k1_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ( ~ v3_struct_0(k1_polyalg1(A))
        & v1_polyalg1(k1_polyalg1(A),A)
        & l1_polyalg1(k1_polyalg1(A),A) ) ) )).

fof(dt_k2_polyalg1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A)
        & ~ v3_struct_0(B)
        & l1_polyalg1(B,A)
        & ~ v1_xboole_0(C)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
     => ( ~ v3_struct_0(k2_polyalg1(A,B,C))
        & v1_polyalg1(k2_polyalg1(A,B,C),A)
        & m1_polyalg1(k2_polyalg1(A,B,C),A,B) ) ) )).

fof(dt_k3_polyalg1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ( ~ v3_struct_0(k3_polyalg1(A))
        & v1_polyalg1(k3_polyalg1(A),A)
        & l1_polyalg1(k3_polyalg1(A),A) ) ) )).

fof(dt_g1_polyalg1,axiom,(
    ! [A,B,C,D,E,F,G] :
      ( ( l1_struct_0(A)
        & v1_funct_1(C)
        & v1_funct_2(C,k2_zfmisc_1(B,B),B)
        & m1_relset_1(C,k2_zfmisc_1(B,B),B)
        & v1_funct_1(D)
        & v1_funct_2(D,k2_zfmisc_1(B,B),B)
        & m1_relset_1(D,k2_zfmisc_1(B,B),B)
        & m1_subset_1(E,B)
        & m1_subset_1(F,B)
        & v1_funct_1(G)
        & v1_funct_2(G,k2_zfmisc_1(u1_struct_0(A),B),B)
        & m1_relset_1(G,k2_zfmisc_1(u1_struct_0(A),B),B) )
     => ( v1_polyalg1(g1_polyalg1(A,B,C,D,E,F,G),A)
        & l1_polyalg1(g1_polyalg1(A,B,C,D,E,F,G),A) ) ) )).

fof(free_g1_polyalg1,axiom,(
    ! [A,B,C,D,E,F,G] :
      ( ( l1_struct_0(A)
        & v1_funct_1(C)
        & v1_funct_2(C,k2_zfmisc_1(B,B),B)
        & m1_relset_1(C,k2_zfmisc_1(B,B),B)
        & v1_funct_1(D)
        & v1_funct_2(D,k2_zfmisc_1(B,B),B)
        & m1_relset_1(D,k2_zfmisc_1(B,B),B)
        & m1_subset_1(E,B)
        & m1_subset_1(F,B)
        & v1_funct_1(G)
        & v1_funct_2(G,k2_zfmisc_1(u1_struct_0(A),B),B)
        & m1_relset_1(G,k2_zfmisc_1(u1_struct_0(A),B),B) )
     => ! [H,I,J,K,L,M,N] :
          ( g1_polyalg1(A,B,C,D,E,F,G) = g1_polyalg1(H,I,J,K,L,M,N)
         => ( A = H
            & B = I
            & C = J
            & D = K
            & E = L
            & F = M
            & G = N ) ) ) )).
%------------------------------------------------------------------------------