SET007 Axioms: SET007+401.ax


%------------------------------------------------------------------------------
% File     : SET007+401 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Quantales
% 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    : quantal1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  127 (   5 unit)
%            Number of atoms       : 1115 ( 103 equality)
%            Maximal formula depth :   21 (   9 average)
%            Number of connectives : 1099 ( 111 ~  ;   0  |; 695  &)
%                                         (  43 <=>; 250 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   56 (   0 propositional; 1-3 arity)
%            Number of functors    :   67 (  13 constant; 0-6 arity)
%            Number of variables   :  367 (   0 singleton; 326 !;  41 ?)
%            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_quantal1,axiom,(
    ? [A] :
      ( l1_quantal1(A)
      & v2_quantal1(A) ) )).

fof(rc2_quantal1,axiom,(
    ? [A] :
      ( l1_quantal1(A)
      & ~ v3_struct_0(A) ) )).

fof(rc3_quantal1,axiom,(
    ? [A] :
      ( l2_quantal1(A)
      & v3_quantal1(A) ) )).

fof(rc4_quantal1,axiom,(
    ? [A] :
      ( l2_quantal1(A)
      & ~ v3_struct_0(A) ) )).

fof(cc1_quantal1,axiom,(
    ! [A] :
      ( l1_group_1(A)
     => ( ( ~ v3_struct_0(A)
          & v6_quantal1(A) )
       => ( ~ v3_struct_0(A)
          & v4_quantal1(A)
          & v5_quantal1(A) ) ) ) )).

fof(cc2_quantal1,axiom,(
    ! [A] :
      ( l1_group_1(A)
     => ( ( ~ v3_struct_0(A)
          & v4_quantal1(A)
          & v5_quantal1(A) )
       => ( ~ v3_struct_0(A)
          & v6_quantal1(A) ) ) ) )).

fof(rc5_quantal1,axiom,(
    ? [A] :
      ( l1_group_1(A)
      & ~ v3_struct_0(A)
      & v4_quantal1(A)
      & v5_quantal1(A)
      & v6_quantal1(A) ) )).

fof(fc1_quantal1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(A,A),A)
        & m1_relset_1(B,k2_zfmisc_1(A,A),A)
        & v1_funct_1(C)
        & v1_funct_2(C,k2_zfmisc_1(A,A),A)
        & m1_relset_1(C,k2_zfmisc_1(A,A),A)
        & v1_funct_1(D)
        & v1_funct_2(D,k2_zfmisc_1(A,A),A)
        & m1_relset_1(D,k2_zfmisc_1(A,A),A) )
     => ( ~ v3_struct_0(g1_quantal1(A,B,C,D))
        & v2_quantal1(g1_quantal1(A,B,C,D)) ) ) )).

fof(rc6_quantal1,axiom,(
    ? [A] :
      ( l1_quantal1(A)
      & ~ v3_struct_0(A)
      & v4_lattices(A)
      & v5_lattices(A)
      & v6_lattices(A)
      & v7_lattices(A)
      & v8_lattices(A)
      & v9_lattices(A)
      & v10_lattices(A)
      & v4_lattice3(A)
      & v2_group_1(A)
      & v4_group_1(A)
      & v7_group_1(A)
      & v4_quantal1(A)
      & v5_quantal1(A)
      & v6_quantal1(A)
      & v7_quantal1(A)
      & v8_quantal1(A) ) )).

fof(fc2_quantal1,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(A,A),A)
        & m1_relset_1(B,k2_zfmisc_1(A,A),A)
        & v1_funct_1(C)
        & v1_funct_2(C,k2_zfmisc_1(A,A),A)
        & m1_relset_1(C,k2_zfmisc_1(A,A),A)
        & v1_funct_1(D)
        & v1_funct_2(D,k2_zfmisc_1(A,A),A)
        & m1_relset_1(D,k2_zfmisc_1(A,A),A)
        & m1_subset_1(E,A) )
     => ( ~ v3_struct_0(g2_quantal1(A,B,C,D,E))
        & v3_quantal1(g2_quantal1(A,B,C,D,E)) ) ) )).

fof(rc7_quantal1,axiom,(
    ? [A] :
      ( l2_quantal1(A)
      & ~ v3_struct_0(A)
      & v4_lattices(A)
      & v5_lattices(A)
      & v6_lattices(A)
      & v7_lattices(A)
      & v8_lattices(A)
      & v9_lattices(A)
      & v10_lattices(A)
      & v4_lattice3(A) ) )).

fof(cc3_quantal1,axiom,(
    ! [A] :
      ( l2_quantal1(A)
     => ( ( ~ v3_struct_0(A)
          & v10_lattices(A)
          & v4_lattice3(A)
          & v7_quantal1(A)
          & v8_quantal1(A) )
       => ( ~ v3_struct_0(A)
          & v4_lattices(A)
          & v5_lattices(A)
          & v6_lattices(A)
          & v7_lattices(A)
          & v8_lattices(A)
          & v9_lattices(A)
          & v10_lattices(A)
          & v4_lattice3(A)
          & v9_quantal1(A)
          & v10_quantal1(A) ) ) ) )).

fof(rc8_quantal1,axiom,(
    ? [A] :
      ( l2_quantal1(A)
      & ~ v3_struct_0(A)
      & v4_lattices(A)
      & v5_lattices(A)
      & v6_lattices(A)
      & v7_lattices(A)
      & v8_lattices(A)
      & v9_lattices(A)
      & v10_lattices(A)
      & v4_lattice3(A)
      & v2_group_1(A)
      & v4_group_1(A)
      & v7_group_1(A)
      & v4_quantal1(A)
      & v5_quantal1(A)
      & v6_quantal1(A)
      & v7_quantal1(A)
      & v8_quantal1(A)
      & v9_quantal1(A)
      & v10_quantal1(A) ) )).

fof(rc9_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & l3_lattices(A) )
     => ? [B] :
          ( m1_relset_1(B,u1_struct_0(A),u1_struct_0(A))
          & v1_relat_1(B)
          & v1_funct_1(B)
          & ~ v1_xboole_0(B)
          & v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
          & v1_partfun1(B,u1_struct_0(A),u1_struct_0(A))
          & v12_quantal1(B,A)
          & v13_quantal1(B,A)
          & v14_quantal1(B,A) ) ) )).

fof(rc10_quantal1,axiom,(
    ? [A] :
      ( l3_quantal1(A)
      & v19_quantal1(A) ) )).

fof(rc11_quantal1,axiom,(
    ? [A] :
      ( l3_quantal1(A)
      & ~ v3_struct_0(A) ) )).

fof(fc3_quantal1,axiom,(
    ! [A,B,C,D,E,F] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(A,A),A)
        & m1_relset_1(B,k2_zfmisc_1(A,A),A)
        & v1_funct_1(C)
        & v1_funct_2(C,k2_zfmisc_1(A,A),A)
        & m1_relset_1(C,k2_zfmisc_1(A,A),A)
        & v1_funct_1(D)
        & v1_funct_2(D,k2_zfmisc_1(A,A),A)
        & m1_relset_1(D,k2_zfmisc_1(A,A),A)
        & m1_subset_1(E,A)
        & m1_subset_1(F,A) )
     => ( ~ v3_struct_0(g3_quantal1(A,B,C,D,E,F))
        & v19_quantal1(g3_quantal1(A,B,C,D,E,F)) ) ) )).

fof(rc12_quantal1,axiom,(
    ? [A] :
      ( l3_quantal1(A)
      & ~ v3_struct_0(A)
      & v4_lattices(A)
      & v5_lattices(A)
      & v6_lattices(A)
      & v7_lattices(A)
      & v8_lattices(A)
      & v9_lattices(A)
      & v10_lattices(A)
      & v4_lattice3(A)
      & v2_group_1(A)
      & v4_group_1(A)
      & v7_group_1(A)
      & v7_quantal1(A)
      & v8_quantal1(A)
      & v9_quantal1(A)
      & v10_quantal1(A)
      & v19_quantal1(A)
      & v20_quantal1(A)
      & v21_quantal1(A) ) )).

fof(t1_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_lattices(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l3_lattices(B) )
         => ( g3_lattices(u1_struct_0(A),u2_lattices(A),u1_lattices(A)) = g3_lattices(u1_struct_0(B),u2_lattices(B),u1_lattices(B))
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [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(B))
                           => ( ( C = E
                                & D = F )
                             => ( k1_lattices(A,C,D) = k1_lattices(B,E,F)
                                & k2_lattices(A,C,D) = k2_lattices(B,E,F)
                                & ( r1_lattices(A,C,D)
                                 => r1_lattices(B,E,F) )
                                & ( r1_lattices(B,E,F)
                                 => r1_lattices(A,C,D) ) ) ) ) ) ) ) ) ) ) )).

fof(t2_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_lattices(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l3_lattices(B) )
         => ( g3_lattices(u1_struct_0(A),u2_lattices(A),u1_lattices(A)) = g3_lattices(u1_struct_0(B),u2_lattices(B),u1_lattices(B))
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(B))
                   => ! [E] :
                        ( C = D
                       => ( ( r3_lattice3(A,C,E)
                           => r3_lattice3(B,D,E) )
                          & ( r3_lattice3(B,D,E)
                           => r3_lattice3(A,C,E) )
                          & ( r4_lattice3(A,C,E)
                           => r4_lattice3(B,D,E) )
                          & ( r4_lattice3(B,D,E)
                           => r4_lattice3(A,C,E) ) ) ) ) ) ) ) ) )).

fof(d1_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v1_quantal1(B,A)
          <=> ! [C] :
                ( ( v1_finset_1(C)
                  & m1_subset_1(C,k1_zfmisc_1(B)) )
               => ? [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                    & r1_lattices(A,k15_lattice3(A,C),D)
                    & r2_hidden(D,B) ) ) ) ) ) )).

fof(t3_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ~ ( v1_quantal1(B,A)
              & v1_xboole_0(B) ) ) ) )).

fof(d2_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v4_quantal1(A)
      <=> ? [B] :
            ( m1_subset_1(B,u1_struct_0(A))
            & ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => k1_group_1(A,B,C) = B ) ) ) ) )).

fof(d3_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v5_quantal1(A)
      <=> ? [B] :
            ( m1_subset_1(B,u1_struct_0(A))
            & ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => k1_group_1(A,C,B) = B ) ) ) ) )).

fof(d4_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ( v6_quantal1(A)
      <=> ( v4_quantal1(A)
          & v5_quantal1(A) ) ) ) )).

fof(d7_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_quantal1(A) )
     => ( v9_quantal1(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))
                   => ( k4_monoid_0(A,k1_lattices(A,B,C),D) = k1_lattices(A,k4_monoid_0(A,B,D),k4_monoid_0(A,C,D))
                      & k4_monoid_0(A,D,k1_lattices(A,B,C)) = k1_lattices(A,k4_monoid_0(A,D,B),k4_monoid_0(A,D,C)) ) ) ) ) ) ) )).

fof(t4_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_quantal1(A) )
     => ( g3_lattices(u1_struct_0(A),u2_lattices(A),u1_lattices(A)) = k1_lattice3(k1_xboole_0)
       => ( v4_group_1(A)
          & v7_group_1(A)
          & v2_group_1(A)
          & v6_quantal1(A)
          & v4_lattice3(A)
          & v7_quantal1(A)
          & v8_quantal1(A)
          & v10_lattices(A) ) ) ) )).

fof(t6_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & v7_quantal1(A)
        & v8_quantal1(A)
        & l1_quantal1(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))
                 => ( k4_monoid_0(A,k3_lattices(A,B,C),D) = k3_lattices(A,k4_monoid_0(A,B,D),k4_monoid_0(A,C,D))
                    & k4_monoid_0(A,D,k3_lattices(A,B,C)) = k3_lattices(A,k4_monoid_0(A,D,B),k4_monoid_0(A,D,C)) ) ) ) ) ) )).

fof(t7_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l2_quantal1(A) )
     => ( ( ~ v3_struct_0(A)
          & v10_lattices(A)
          & v4_lattice3(A)
          & v4_group_1(A)
          & v7_quantal1(A)
          & v8_quantal1(A)
          & l1_quantal1(A) )
       => ( ~ v3_struct_0(A)
          & v10_lattices(A)
          & v4_lattice3(A)
          & v2_group_1(A)
          & v4_group_1(A)
          & v4_quantal1(A)
          & v6_quantal1(A)
          & v9_quantal1(A)
          & v10_quantal1(A)
          & l2_quantal1(A) ) ) ) )).

fof(t8_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & v4_group_1(A)
        & v7_quantal1(A)
        & v8_quantal1(A)
        & l1_quantal1(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))
                 => ( r3_lattices(A,B,C)
                   => ( r3_lattices(A,k4_monoid_0(A,B,D),k4_monoid_0(A,C,D))
                      & r3_lattices(A,k4_monoid_0(A,D,B),k4_monoid_0(A,D,C)) ) ) ) ) ) ) )).

fof(t9_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & v4_group_1(A)
        & v7_quantal1(A)
        & v8_quantal1(A)
        & l1_quantal1(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))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(A))
                     => ( ( r3_lattices(A,B,C)
                          & r3_lattices(A,D,E) )
                       => r3_lattices(A,k4_monoid_0(A,B,D),k4_monoid_0(A,C,E)) ) ) ) ) ) ) )).

fof(d9_quantal1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v11_quantal1(A)
      <=> k5_relat_1(A,A) = A ) ) )).

fof(d10_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_lattices(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
            & m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
         => ( v12_quantal1(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => r1_lattices(A,C,k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,C)) ) ) ) ) )).

fof(d11_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_lattices(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
            & m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
         => ( v13_quantal1(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => r1_lattices(A,k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,C),C) ) ) ) ) )).

fof(d12_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_lattices(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
            & m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
         => ( v14_quantal1(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ( r1_lattices(A,C,D)
                     => r1_lattices(A,k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,C),k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,D)) ) ) ) ) ) ) )).

fof(d14_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_quantal1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
            & m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
         => ( v16_quantal1(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => r1_lattices(A,k4_monoid_0(A,k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,C),k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,D)),k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,k4_monoid_0(A,C,D))) ) ) ) ) ) )).

fof(t11_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & v4_group_1(A)
        & v7_quantal1(A)
        & v8_quantal1(A)
        & l1_quantal1(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))
                 => ( r3_lattices(A,k4_monoid_0(A,B,C),D)
                  <=> r3_lattices(A,C,k2_quantal1(A,B,D)) ) ) ) ) ) )).

fof(t12_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & v4_group_1(A)
        & v7_quantal1(A)
        & v8_quantal1(A)
        & l1_quantal1(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))
                 => ( r3_lattices(A,k4_monoid_0(A,B,C),D)
                  <=> r3_lattices(A,B,k1_quantal1(A,C,D)) ) ) ) ) ) )).

fof(t13_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & v4_group_1(A)
        & v7_quantal1(A)
        & v8_quantal1(A)
        & l1_quantal1(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))
                 => ( r3_lattices(A,C,D)
                   => ( r3_lattices(A,k1_quantal1(A,D,B),k1_quantal1(A,C,B))
                      & r3_lattices(A,k2_quantal1(A,D,B),k2_quantal1(A,C,B)) ) ) ) ) ) ) )).

fof(t14_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & v4_group_1(A)
        & v7_quantal1(A)
        & v8_quantal1(A)
        & l1_quantal1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(A))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(A)) )
             => ( ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => k8_funct_2(u1_struct_0(A),u1_struct_0(A),C,D) = k1_quantal1(A,k1_quantal1(A,D,B),B) )
               => v14_quantal1(C,A) ) ) ) ) )).

fof(d17_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_quantal1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( v17_quantal1(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ( k2_quantal1(A,k1_quantal1(A,C,B),B) = C
                  & k1_quantal1(A,k2_quantal1(A,C,B),B) = C ) ) ) ) ) )).

fof(d18_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_quantal1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( v18_quantal1(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => k1_quantal1(A,C,B) = k2_quantal1(A,C,B) ) ) ) ) )).

fof(t15_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & v4_group_1(A)
        & v7_quantal1(A)
        & v8_quantal1(A)
        & l1_quantal1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( v18_quantal1(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ( r3_lattices(A,k4_monoid_0(A,C,D),B)
                     => r3_lattices(A,k4_monoid_0(A,D,C),B) ) ) ) ) ) ) )).

fof(t16_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & v4_group_1(A)
        & v7_quantal1(A)
        & v8_quantal1(A)
        & l1_quantal1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( v18_quantal1(B,A)
               => ( r3_lattices(A,C,k1_quantal1(A,k1_quantal1(A,C,B),B))
                  & r3_lattices(A,C,k2_quantal1(A,k2_quantal1(A,C,B),B)) ) ) ) ) ) )).

fof(t17_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & v4_group_1(A)
        & v7_quantal1(A)
        & v8_quantal1(A)
        & l1_quantal1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( v18_quantal1(B,A)
               => ( k1_quantal1(A,C,B) = k1_quantal1(A,k1_quantal1(A,k1_quantal1(A,C,B),B),B)
                  & k2_quantal1(A,C,B) = k2_quantal1(A,k2_quantal1(A,k2_quantal1(A,C,B),B),B) ) ) ) ) ) )).

fof(t18_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & v4_group_1(A)
        & v7_quantal1(A)
        & v8_quantal1(A)
        & l1_quantal1(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))
                 => ( v18_quantal1(B,A)
                   => r3_lattices(A,k4_monoid_0(A,k1_quantal1(A,k1_quantal1(A,C,B),B),k1_quantal1(A,k1_quantal1(A,D,B),B)),k1_quantal1(A,k1_quantal1(A,k4_monoid_0(A,C,D),B),B)) ) ) ) ) ) )).

fof(t19_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & v4_group_1(A)
        & v7_quantal1(A)
        & v8_quantal1(A)
        & l1_quantal1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( v17_quantal1(B,A)
           => ( v2_group_1(A)
              & k3_binop_1(u1_struct_0(A),u1_group_1(A)) = k1_quantal1(A,B,B)
              & k3_binop_1(u1_struct_0(A),u1_group_1(A)) = k2_quantal1(A,B,B) ) ) ) ) )).

fof(t20_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & v4_group_1(A)
        & v7_quantal1(A)
        & v8_quantal1(A)
        & l1_quantal1(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))
                 => ( v17_quantal1(B,A)
                   => ( k1_quantal1(A,C,D) = k1_quantal1(A,k4_monoid_0(A,C,k2_quantal1(A,D,B)),B)
                      & k2_quantal1(A,C,D) = k2_quantal1(A,k4_monoid_0(A,k1_quantal1(A,D,B),C),B) ) ) ) ) ) ) )).

fof(d19_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_quantal1(A) )
     => ( v20_quantal1(A)
      <=> v18_quantal1(u1_quantal1(A),A) ) ) )).

fof(d20_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_quantal1(A) )
     => ( v21_quantal1(A)
      <=> v17_quantal1(u1_quantal1(A),A) ) ) )).

fof(t21_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_quantal1(A) )
     => ( g3_lattices(u1_struct_0(A),u2_lattices(A),u1_lattices(A)) = k1_lattice3(k1_xboole_0)
       => ( v20_quantal1(A)
          & v21_quantal1(A) ) ) ) )).

fof(d21_quantal1,axiom,(
    ! [A] :
      ( l3_quantal1(A)
     => k3_quantal1(A) = u1_quantal1(A) ) )).

fof(d22_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_quantal1(A) )
     => k4_quantal1(A) = k1_quantal1(A,k3_quantal1(A),k3_quantal1(A)) ) )).

fof(d23_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_quantal1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k5_quantal1(A,B) = k1_quantal1(A,B,k3_quantal1(A)) ) ) )).

fof(d24_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_quantal1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
            & m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
         => ( B = k6_quantal1(A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,C) = k5_quantal1(A,C) ) ) ) ) )).

fof(d25_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_quantal1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
            & m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
         => k7_quantal1(A,B) = k2_monoid_0(u1_struct_0(A),B,k6_quantal1(A)) ) ) )).

fof(d26_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_quantal1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A))
            & m2_relset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) )
         => k8_quantal1(A,B) = k7_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A),u1_struct_0(A),B,k6_quantal1(A)) ) ) )).

fof(t22_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_quantal1(A)
        & v8_quantal1(A)
        & v20_quantal1(A)
        & v21_quantal1(A)
        & l3_quantal1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k5_quantal1(A,k5_quantal1(A,B)) = B ) ) )).

fof(t23_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_quantal1(A)
        & v8_quantal1(A)
        & v20_quantal1(A)
        & v21_quantal1(A)
        & l3_quantal1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r3_lattices(A,B,C)
               => r3_lattices(A,k5_quantal1(A,C),k5_quantal1(A,B)) ) ) ) ) )).

fof(t26_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_quantal1(A)
        & v8_quantal1(A)
        & v20_quantal1(A)
        & v21_quantal1(A)
        & l3_quantal1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( k5_quantal1(A,k3_lattices(A,B,C)) = k4_lattices(A,k5_quantal1(A,B),k5_quantal1(A,C))
                & k5_quantal1(A,k4_lattices(A,B,C)) = k3_lattices(A,k5_quantal1(A,B),k5_quantal1(A,C)) ) ) ) ) )).

fof(d27_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_quantal1(A)
        & v8_quantal1(A)
        & v20_quantal1(A)
        & v21_quantal1(A)
        & l3_quantal1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k9_quantal1(A,B,C) = k5_quantal1(A,k4_monoid_0(A,k5_quantal1(A,B),k5_quantal1(A,C))) ) ) ) )).

fof(t29_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_quantal1(A)
        & v8_quantal1(A)
        & v20_quantal1(A)
        & v21_quantal1(A)
        & l3_quantal1(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))
                 => ( k9_quantal1(A,B,k4_lattices(A,C,D)) = k4_lattices(A,k9_quantal1(A,B,C),k9_quantal1(A,B,D))
                    & k9_quantal1(A,k4_lattices(A,C,D),B) = k4_lattices(A,k9_quantal1(A,C,B),k9_quantal1(A,D,B)) ) ) ) ) ) )).

fof(t30_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_quantal1(A)
        & v8_quantal1(A)
        & v20_quantal1(A)
        & v21_quantal1(A)
        & l3_quantal1(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))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(A))
                     => ( ( r3_lattices(A,B,C)
                          & r3_lattices(A,D,E) )
                       => r3_lattices(A,k9_quantal1(A,B,D),k9_quantal1(A,C,E)) ) ) ) ) ) ) )).

fof(t31_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_quantal1(A)
        & v8_quantal1(A)
        & v20_quantal1(A)
        & v21_quantal1(A)
        & l3_quantal1(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))
                 => k9_quantal1(A,k9_quantal1(A,B,C),D) = k9_quantal1(A,B,k9_quantal1(A,C,D)) ) ) ) ) )).

fof(t32_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_quantal1(A)
        & v8_quantal1(A)
        & v20_quantal1(A)
        & v21_quantal1(A)
        & l3_quantal1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( k4_monoid_0(A,B,k4_quantal1(A)) = B
            & k4_monoid_0(A,k4_quantal1(A),B) = B ) ) ) )).

fof(t33_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_quantal1(A)
        & v8_quantal1(A)
        & v20_quantal1(A)
        & v21_quantal1(A)
        & l3_quantal1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( k9_quantal1(A,B,k3_quantal1(A)) = B
            & k9_quantal1(A,k3_quantal1(A),B) = B ) ) ) )).

fof(t34_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & v4_group_1(A)
        & v7_quantal1(A)
        & v8_quantal1(A)
        & l1_quantal1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
            & m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
         => ~ ( v14_quantal1(B,A)
              & v11_quantal1(B)
              & v15_quantal1(B,A)
              & ! [C] :
                  ( ( ~ v3_struct_0(C)
                    & v10_lattices(C)
                    & v4_lattice3(C)
                    & l3_lattices(C) )
                 => ~ ( u1_struct_0(C) = k2_relat_1(B)
                      & ! [D] :
                          ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(C)))
                         => k15_lattice3(C,D) = k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,k15_lattice3(A,D)) ) ) ) ) ) ) )).

fof(dt_l1_quantal1,axiom,(
    ! [A] :
      ( l1_quantal1(A)
     => ( l3_lattices(A)
        & l1_group_1(A) ) ) )).

fof(existence_l1_quantal1,axiom,(
    ? [A] : l1_quantal1(A) )).

fof(dt_l2_quantal1,axiom,(
    ! [A] :
      ( l2_quantal1(A)
     => ( l1_quantal1(A)
        & l1_vectsp_1(A) ) ) )).

fof(existence_l2_quantal1,axiom,(
    ? [A] : l2_quantal1(A) )).

fof(dt_l3_quantal1,axiom,(
    ! [A] :
      ( l3_quantal1(A)
     => l2_quantal1(A) ) )).

fof(existence_l3_quantal1,axiom,(
    ? [A] : l3_quantal1(A) )).

fof(abstractness_v2_quantal1,axiom,(
    ! [A] :
      ( l1_quantal1(A)
     => ( v2_quantal1(A)
       => A = g1_quantal1(u1_struct_0(A),u2_lattices(A),u1_lattices(A),u1_group_1(A)) ) ) )).

fof(abstractness_v3_quantal1,axiom,(
    ! [A] :
      ( l2_quantal1(A)
     => ( v3_quantal1(A)
       => A = g2_quantal1(u1_struct_0(A),u2_lattices(A),u1_lattices(A),u1_group_1(A),u1_vectsp_1(A)) ) ) )).

fof(abstractness_v19_quantal1,axiom,(
    ! [A] :
      ( l3_quantal1(A)
     => ( v19_quantal1(A)
       => A = g3_quantal1(u1_struct_0(A),u2_lattices(A),u1_lattices(A),u1_group_1(A),u1_vectsp_1(A),u1_quantal1(A)) ) ) )).

fof(dt_k1_quantal1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_quantal1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => m1_subset_1(k1_quantal1(A,B,C),u1_struct_0(A)) ) )).

fof(dt_k2_quantal1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_quantal1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => m1_subset_1(k2_quantal1(A,B,C),u1_struct_0(A)) ) )).

fof(dt_k3_quantal1,axiom,(
    ! [A] :
      ( l3_quantal1(A)
     => m1_subset_1(k3_quantal1(A),u1_struct_0(A)) ) )).

fof(dt_k4_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_quantal1(A) )
     => m1_subset_1(k4_quantal1(A),u1_struct_0(A)) ) )).

fof(dt_k5_quantal1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l3_quantal1(A)
        & m1_subset_1(B,u1_struct_0(A)) )
     => m1_subset_1(k5_quantal1(A,B),u1_struct_0(A)) ) )).

fof(dt_k6_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_quantal1(A) )
     => ( v1_funct_1(k6_quantal1(A))
        & v1_funct_2(k6_quantal1(A),u1_struct_0(A),u1_struct_0(A))
        & m2_relset_1(k6_quantal1(A),u1_struct_0(A),u1_struct_0(A)) ) ) )).

fof(dt_k7_quantal1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l3_quantal1(A)
        & v1_funct_1(B)
        & v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
        & m1_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
     => ( v1_funct_1(k7_quantal1(A,B))
        & v1_funct_2(k7_quantal1(A,B),u1_struct_0(A),u1_struct_0(A))
        & m2_relset_1(k7_quantal1(A,B),u1_struct_0(A),u1_struct_0(A)) ) ) )).

fof(dt_k8_quantal1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l3_quantal1(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A))
        & m1_relset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) )
     => ( v1_funct_1(k8_quantal1(A,B))
        & v1_funct_2(k8_quantal1(A,B),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A))
        & m2_relset_1(k8_quantal1(A,B),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) ) ) )).

fof(dt_k9_quantal1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_quantal1(A)
        & v8_quantal1(A)
        & v20_quantal1(A)
        & v21_quantal1(A)
        & l3_quantal1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => m1_subset_1(k9_quantal1(A,B,C),u1_struct_0(A)) ) )).

fof(dt_u1_quantal1,axiom,(
    ! [A] :
      ( l3_quantal1(A)
     => m1_subset_1(u1_quantal1(A),u1_struct_0(A)) ) )).

fof(dt_g1_quantal1,axiom,(
    ! [A,B,C,D] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(A,A),A)
        & m1_relset_1(B,k2_zfmisc_1(A,A),A)
        & v1_funct_1(C)
        & v1_funct_2(C,k2_zfmisc_1(A,A),A)
        & m1_relset_1(C,k2_zfmisc_1(A,A),A)
        & v1_funct_1(D)
        & v1_funct_2(D,k2_zfmisc_1(A,A),A)
        & m1_relset_1(D,k2_zfmisc_1(A,A),A) )
     => ( v2_quantal1(g1_quantal1(A,B,C,D))
        & l1_quantal1(g1_quantal1(A,B,C,D)) ) ) )).

fof(free_g1_quantal1,axiom,(
    ! [A,B,C,D] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(A,A),A)
        & m1_relset_1(B,k2_zfmisc_1(A,A),A)
        & v1_funct_1(C)
        & v1_funct_2(C,k2_zfmisc_1(A,A),A)
        & m1_relset_1(C,k2_zfmisc_1(A,A),A)
        & v1_funct_1(D)
        & v1_funct_2(D,k2_zfmisc_1(A,A),A)
        & m1_relset_1(D,k2_zfmisc_1(A,A),A) )
     => ! [E,F,G,H] :
          ( g1_quantal1(A,B,C,D) = g1_quantal1(E,F,G,H)
         => ( A = E
            & B = F
            & C = G
            & D = H ) ) ) )).

fof(dt_g2_quantal1,axiom,(
    ! [A,B,C,D,E] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(A,A),A)
        & m1_relset_1(B,k2_zfmisc_1(A,A),A)
        & v1_funct_1(C)
        & v1_funct_2(C,k2_zfmisc_1(A,A),A)
        & m1_relset_1(C,k2_zfmisc_1(A,A),A)
        & v1_funct_1(D)
        & v1_funct_2(D,k2_zfmisc_1(A,A),A)
        & m1_relset_1(D,k2_zfmisc_1(A,A),A)
        & m1_subset_1(E,A) )
     => ( v3_quantal1(g2_quantal1(A,B,C,D,E))
        & l2_quantal1(g2_quantal1(A,B,C,D,E)) ) ) )).

fof(free_g2_quantal1,axiom,(
    ! [A,B,C,D,E] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(A,A),A)
        & m1_relset_1(B,k2_zfmisc_1(A,A),A)
        & v1_funct_1(C)
        & v1_funct_2(C,k2_zfmisc_1(A,A),A)
        & m1_relset_1(C,k2_zfmisc_1(A,A),A)
        & v1_funct_1(D)
        & v1_funct_2(D,k2_zfmisc_1(A,A),A)
        & m1_relset_1(D,k2_zfmisc_1(A,A),A)
        & m1_subset_1(E,A) )
     => ! [F,G,H,I,J] :
          ( g2_quantal1(A,B,C,D,E) = g2_quantal1(F,G,H,I,J)
         => ( A = F
            & B = G
            & C = H
            & D = I
            & E = J ) ) ) )).

fof(dt_g3_quantal1,axiom,(
    ! [A,B,C,D,E,F] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(A,A),A)
        & m1_relset_1(B,k2_zfmisc_1(A,A),A)
        & v1_funct_1(C)
        & v1_funct_2(C,k2_zfmisc_1(A,A),A)
        & m1_relset_1(C,k2_zfmisc_1(A,A),A)
        & v1_funct_1(D)
        & v1_funct_2(D,k2_zfmisc_1(A,A),A)
        & m1_relset_1(D,k2_zfmisc_1(A,A),A)
        & m1_subset_1(E,A)
        & m1_subset_1(F,A) )
     => ( v19_quantal1(g3_quantal1(A,B,C,D,E,F))
        & l3_quantal1(g3_quantal1(A,B,C,D,E,F)) ) ) )).

fof(free_g3_quantal1,axiom,(
    ! [A,B,C,D,E,F] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(A,A),A)
        & m1_relset_1(B,k2_zfmisc_1(A,A),A)
        & v1_funct_1(C)
        & v1_funct_2(C,k2_zfmisc_1(A,A),A)
        & m1_relset_1(C,k2_zfmisc_1(A,A),A)
        & v1_funct_1(D)
        & v1_funct_2(D,k2_zfmisc_1(A,A),A)
        & m1_relset_1(D,k2_zfmisc_1(A,A),A)
        & m1_subset_1(E,A)
        & m1_subset_1(F,A) )
     => ! [G,H,I,J,K,L] :
          ( g3_quantal1(A,B,C,D,E,F) = g3_quantal1(G,H,I,J,K,L)
         => ( A = G
            & B = H
            & C = I
            & D = J
            & E = K
            & F = L ) ) ) )).

fof(d5_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_quantal1(A) )
     => ( v7_quantal1(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] : k4_monoid_0(A,B,k15_lattice3(A,C)) = k15_lattice3(A,a_3_0_quantal1(A,B,C)) ) ) ) )).

fof(d6_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_quantal1(A) )
     => ( v8_quantal1(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] : k4_monoid_0(A,k15_lattice3(A,C),B) = k15_lattice3(A,a_3_1_quantal1(A,B,C)) ) ) ) )).

fof(d8_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_quantal1(A) )
     => ( v10_quantal1(A)
      <=> ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
           => ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
               => ( ( v1_quantal1(B,A)
                    & v1_quantal1(C,A) )
                 => k4_monoid_0(A,k15_lattice3(A,B),k15_lattice3(A,C)) = k15_lattice3(A,a_3_2_quantal1(A,B,C)) ) ) ) ) ) )).

fof(t5_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & v7_quantal1(A)
        & v8_quantal1(A)
        & l1_quantal1(A) )
     => ! [B,C] : k4_monoid_0(A,k15_lattice3(A,B),k15_lattice3(A,C)) = k15_lattice3(A,a_3_3_quantal1(A,B,C)) ) )).

fof(d13_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_lattices(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
            & m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
         => ( v15_quantal1(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
               => r1_lattices(A,k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,k15_lattice3(A,C)),k15_lattice3(A,a_3_4_quantal1(A,B,C))) ) ) ) ) )).

fof(t10_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & l3_lattices(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
            & m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
         => ( v14_quantal1(B,A)
           => ( v15_quantal1(B,A)
            <=> ! [C] :
                  ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
                 => k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,k15_lattice3(A,C)) = k15_lattice3(A,a_3_5_quantal1(A,B,C)) ) ) ) ) ) )).

fof(d15_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_quantal1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k1_quantal1(A,B,C) = k15_lattice3(A,a_3_6_quantal1(A,B,C)) ) ) ) )).

fof(d16_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_quantal1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k2_quantal1(A,B,C) = k15_lattice3(A,a_3_7_quantal1(A,B,C)) ) ) ) )).

fof(t24_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_quantal1(A)
        & v8_quantal1(A)
        & v20_quantal1(A)
        & v21_quantal1(A)
        & l3_quantal1(A) )
     => ! [B] : k5_quantal1(A,k15_lattice3(A,B)) = k16_lattice3(A,a_2_0_quantal1(A,B)) ) )).

fof(t25_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_quantal1(A)
        & v8_quantal1(A)
        & v20_quantal1(A)
        & v21_quantal1(A)
        & l3_quantal1(A) )
     => ! [B] : k5_quantal1(A,k16_lattice3(A,B)) = k15_lattice3(A,a_2_0_quantal1(A,B)) ) )).

fof(t27_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_quantal1(A)
        & v8_quantal1(A)
        & v20_quantal1(A)
        & v21_quantal1(A)
        & l3_quantal1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( k4_monoid_0(A,B,k15_lattice3(A,C)) = k15_lattice3(A,a_3_8_quantal1(A,B,C))
              & k9_quantal1(A,B,k16_lattice3(A,C)) = k16_lattice3(A,a_3_9_quantal1(A,B,C)) ) ) ) )).

fof(t28_quantal1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_quantal1(A)
        & v8_quantal1(A)
        & v20_quantal1(A)
        & v21_quantal1(A)
        & l3_quantal1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( k4_monoid_0(A,k15_lattice3(A,C),B) = k15_lattice3(A,a_3_10_quantal1(A,B,C))
              & k9_quantal1(A,k16_lattice3(A,C),B) = k16_lattice3(A,a_3_11_quantal1(A,B,C)) ) ) ) )).

fof(s1_quantal1,axiom,(
    a_0_0_quantal1 = a_0_2_quantal1 )).

fof(s2_quantal1,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s2_quantal1)
       => ~ p1_s2_quantal1(A) )
   => a_0_3_quantal1 = k1_xboole_0 )).

fof(s3_quantal1,axiom,(
    k15_lattice3(f1_s3_quantal1,a_0_4_quantal1) = k15_lattice3(f1_s3_quantal1,a_0_5_quantal1) )).

fof(fraenkel_a_3_0_quantal1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & l1_quantal1(B)
        & m1_subset_1(C,u1_struct_0(B)) )
     => ( r2_hidden(A,a_3_0_quantal1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(B))
            & A = k4_monoid_0(B,C,E)
            & r2_hidden(E,D) ) ) ) )).

fof(fraenkel_a_3_1_quantal1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & l1_quantal1(B)
        & m1_subset_1(C,u1_struct_0(B)) )
     => ( r2_hidden(A,a_3_1_quantal1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(B))
            & A = k4_monoid_0(B,E,C)
            & r2_hidden(E,D) ) ) ) )).

fof(fraenkel_a_3_2_quantal1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & l1_quantal1(B)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
        & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
     => ( r2_hidden(A,a_3_2_quantal1(B,C,D))
      <=> ? [E,F] :
            ( m1_subset_1(E,u1_struct_0(B))
            & m1_subset_1(F,u1_struct_0(B))
            & A = k4_monoid_0(B,E,F)
            & r2_hidden(E,C)
            & r2_hidden(F,D) ) ) ) )).

fof(fraenkel_a_3_3_quantal1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & v10_lattices(B)
        & v4_lattice3(B)
        & v7_quantal1(B)
        & v8_quantal1(B)
        & l1_quantal1(B) )
     => ( r2_hidden(A,a_3_3_quantal1(B,C,D))
      <=> ? [E,F] :
            ( m1_subset_1(E,u1_struct_0(B))
            & m1_subset_1(F,u1_struct_0(B))
            & A = k4_monoid_0(B,E,F)
            & r2_hidden(E,C)
            & r2_hidden(F,D) ) ) ) )).

fof(fraenkel_a_3_4_quantal1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & l3_lattices(B)
        & v1_funct_1(C)
        & v1_funct_2(C,u1_struct_0(B),u1_struct_0(B))
        & m2_relset_1(C,u1_struct_0(B),u1_struct_0(B))
        & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
     => ( r2_hidden(A,a_3_4_quantal1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(B))
            & A = k8_funct_2(u1_struct_0(B),u1_struct_0(B),C,E)
            & r2_hidden(E,D) ) ) ) )).

fof(fraenkel_a_3_5_quantal1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & v10_lattices(B)
        & v4_lattice3(B)
        & l3_lattices(B)
        & v1_funct_1(C)
        & v1_funct_2(C,u1_struct_0(B),u1_struct_0(B))
        & m2_relset_1(C,u1_struct_0(B),u1_struct_0(B))
        & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
     => ( r2_hidden(A,a_3_5_quantal1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(B))
            & A = k8_funct_2(u1_struct_0(B),u1_struct_0(B),C,E)
            & r2_hidden(E,D) ) ) ) )).

fof(fraenkel_a_3_6_quantal1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & l1_quantal1(B)
        & m1_subset_1(C,u1_struct_0(B))
        & m1_subset_1(D,u1_struct_0(B)) )
     => ( r2_hidden(A,a_3_6_quantal1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(B))
            & A = E
            & r1_lattices(B,k4_monoid_0(B,E,C),D) ) ) ) )).

fof(fraenkel_a_3_7_quantal1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & l1_quantal1(B)
        & m1_subset_1(C,u1_struct_0(B))
        & m1_subset_1(D,u1_struct_0(B)) )
     => ( r2_hidden(A,a_3_7_quantal1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(B))
            & A = E
            & r1_lattices(B,k4_monoid_0(B,C,E),D) ) ) ) )).

fof(fraenkel_a_2_0_quantal1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & v10_lattices(B)
        & v4_lattice3(B)
        & v2_group_1(B)
        & v4_group_1(B)
        & v7_quantal1(B)
        & v8_quantal1(B)
        & v20_quantal1(B)
        & v21_quantal1(B)
        & l3_quantal1(B) )
     => ( r2_hidden(A,a_2_0_quantal1(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,u1_struct_0(B))
            & A = k5_quantal1(B,D)
            & r2_hidden(D,C) ) ) ) )).

fof(fraenkel_a_3_8_quantal1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & v10_lattices(B)
        & v4_lattice3(B)
        & v2_group_1(B)
        & v4_group_1(B)
        & v7_quantal1(B)
        & v8_quantal1(B)
        & v20_quantal1(B)
        & v21_quantal1(B)
        & l3_quantal1(B)
        & m1_subset_1(C,u1_struct_0(B)) )
     => ( r2_hidden(A,a_3_8_quantal1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(B))
            & A = k4_monoid_0(B,C,E)
            & r2_hidden(E,D) ) ) ) )).

fof(fraenkel_a_3_9_quantal1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & v10_lattices(B)
        & v4_lattice3(B)
        & v2_group_1(B)
        & v4_group_1(B)
        & v7_quantal1(B)
        & v8_quantal1(B)
        & v20_quantal1(B)
        & v21_quantal1(B)
        & l3_quantal1(B)
        & m1_subset_1(C,u1_struct_0(B)) )
     => ( r2_hidden(A,a_3_9_quantal1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(B))
            & A = k9_quantal1(B,C,E)
            & r2_hidden(E,D) ) ) ) )).

fof(fraenkel_a_3_10_quantal1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & v10_lattices(B)
        & v4_lattice3(B)
        & v2_group_1(B)
        & v4_group_1(B)
        & v7_quantal1(B)
        & v8_quantal1(B)
        & v20_quantal1(B)
        & v21_quantal1(B)
        & l3_quantal1(B)
        & m1_subset_1(C,u1_struct_0(B)) )
     => ( r2_hidden(A,a_3_10_quantal1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(B))
            & A = k4_monoid_0(B,E,C)
            & r2_hidden(E,D) ) ) ) )).

fof(fraenkel_a_3_11_quantal1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & v10_lattices(B)
        & v4_lattice3(B)
        & v2_group_1(B)
        & v4_group_1(B)
        & v7_quantal1(B)
        & v8_quantal1(B)
        & v20_quantal1(B)
        & v21_quantal1(B)
        & l3_quantal1(B)
        & m1_subset_1(C,u1_struct_0(B)) )
     => ( r2_hidden(A,a_3_11_quantal1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(B))
            & A = k9_quantal1(B,E,C)
            & r2_hidden(E,D) ) ) ) )).

fof(fraenkel_a_0_0_quantal1,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_0_quantal1)
    <=> ? [B] :
          ( m1_subset_1(B,f2_s1_quantal1)
          & A = f3_s1_quantal1(B)
          & r2_hidden(B,a_0_1_quantal1) ) ) )).

fof(fraenkel_a_0_1_quantal1,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_1_quantal1)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s1_quantal1)
          & A = f4_s1_quantal1(B)
          & p1_s1_quantal1(B) ) ) )).

fof(fraenkel_a_0_2_quantal1,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_2_quantal1)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s1_quantal1)
          & A = f3_s1_quantal1(f4_s1_quantal1(B))
          & p1_s1_quantal1(B) ) ) )).

fof(fraenkel_a_0_3_quantal1,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_3_quantal1)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s2_quantal1)
          & A = f2_s2_quantal1(B)
          & p1_s2_quantal1(B) ) ) )).

fof(fraenkel_a_0_4_quantal1,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_4_quantal1)
    <=> ? [B] :
          ( m1_subset_1(B,u1_struct_0(f1_s3_quantal1))
          & A = k15_lattice3(f1_s3_quantal1,a_1_0_quantal1(B))
          & r2_hidden(B,f3_s3_quantal1) ) ) )).

fof(fraenkel_a_1_0_quantal1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,u1_struct_0(f1_s3_quantal1))
     => ( r2_hidden(A,a_1_0_quantal1(B))
      <=> ? [C] :
            ( m1_subset_1(C,u1_struct_0(f1_s3_quantal1))
            & A = f2_s3_quantal1(B,C)
            & r2_hidden(C,f4_s3_quantal1) ) ) ) )).

fof(fraenkel_a_0_5_quantal1,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_5_quantal1)
    <=> ? [B,C] :
          ( m1_subset_1(B,u1_struct_0(f1_s3_quantal1))
          & m1_subset_1(C,u1_struct_0(f1_s3_quantal1))
          & A = f2_s3_quantal1(B,C)
          & r2_hidden(B,f3_s3_quantal1)
          & r2_hidden(C,f4_s3_quantal1) ) ) )).
%------------------------------------------------------------------------------