% Mizar problem: t22_quantal1,quantal1,1122,14 
fof(t22_quantal1,conjecture,(
    ! [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 ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc1_finset_1,cc1_lattices,cc2_lattices,cc3_quantal1,d15_quantal1,d16_quantal1,d17_quantal1,d18_quantal1,d19_quantal1,d20_quantal1,d21_quantal1,d23_quantal1,dt_k15_lattice3,dt_k1_group_1,dt_k1_quantal1,dt_k1_xboole_0,dt_k2_quantal1,dt_k3_quantal1,dt_k4_monoid_0,dt_k5_quantal1,dt_l1_group_1,dt_l1_lattices,dt_l1_quantal1,dt_l1_struct_0,dt_l1_vectsp_1,dt_l2_lattices,dt_l2_quantal1,dt_l3_lattices,dt_l3_quantal1,dt_m1_subset_1,dt_u1_quantal1,dt_u1_struct_0,existence_l1_group_1,existence_l1_lattices,existence_l1_quantal1,existence_l1_struct_0,existence_l1_vectsp_1,existence_l2_lattices,existence_l2_quantal1,existence_l3_lattices,existence_l3_quantal1,existence_m1_subset_1,fc1_struct_0,fc1_xboole_0,fraenkel_a_3_10_quantal1,fraenkel_a_3_11_quantal1,rc11_quantal1,rc1_finset_1,rc1_xboole_0,rc2_quantal1,rc2_xboole_0,rc3_struct_0,rc4_quantal1,rc7_quantal1,redefinition_k4_monoid_0,t1_subset,t2_subset,t2_tarski,t6_boole,t7_boole,t8_boole]),
    [file(quantal1,t22_quantal1)]).

fof(antisymmetry_r2_hidden,axiom,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => ~ r2_hidden(B,A) ) ),
    file(hidden,r2_hidden),
    []).

fof(cc1_finset_1,axiom,(
    ! [A] : 
      ( v1_xboole_0(A)
     => v1_finset_1(A) ) ),
    file(finset_1,cc1_finset_1),
    []).

fof(cc1_lattices,axiom,(
    ! [A] : 
      ( l3_lattices(A)
     => ( ( ~ v3_struct_0(A)
          & v10_lattices(A) )
       => ( ~ v3_struct_0(A)
          & v4_lattices(A)
          & v5_lattices(A)
          & v6_lattices(A)
          & v7_lattices(A)
          & v8_lattices(A)
          & v9_lattices(A) ) ) ) ),
    file(lattices,cc1_lattices),
    []).

fof(cc2_lattices,axiom,(
    ! [A] : 
      ( l3_lattices(A)
     => ( ( ~ v3_struct_0(A)
          & v4_lattices(A)
          & v5_lattices(A)
          & v6_lattices(A)
          & v7_lattices(A)
          & v8_lattices(A)
          & v9_lattices(A) )
       => ( ~ v3_struct_0(A)
          & v10_lattices(A) ) ) ) ),
    file(lattices,cc2_lattices),
    []).

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) ) ) ) ),
    file(quantal1,cc3_quantal1),
    []).

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_10_quantal1(A,B,C)) ) ) ) ),
    file(quantal1,d15_quantal1),
    []).

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_11_quantal1(A,B,C)) ) ) ) ),
    file(quantal1,d16_quantal1),
    []).

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 ) ) ) ) ) ),
    file(quantal1,d17_quantal1),
    []).

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) ) ) ) ) ),
    file(quantal1,d18_quantal1),
    []).

fof(d19_quantal1,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l3_quantal1(A) )
     => ( v20_quantal1(A)
      <=> v18_quantal1(u1_quantal1(A),A) ) ) ),
    file(quantal1,d19_quantal1),
    []).

fof(d20_quantal1,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l3_quantal1(A) )
     => ( v21_quantal1(A)
      <=> v17_quantal1(u1_quantal1(A),A) ) ) ),
    file(quantal1,d20_quantal1),
    []).

fof(d21_quantal1,axiom,(
    ! [A] : 
      ( l3_quantal1(A)
     => k3_quantal1(A) = u1_quantal1(A) ) ),
    file(quantal1,d21_quantal1),
    []).

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)) ) ) ),
    file(quantal1,d23_quantal1),
    []).

fof(dt_k15_lattice3,axiom,(
    ! [A,B] : 
      ( ( ~ v3_struct_0(A)
        & l3_lattices(A) )
     => m1_subset_1(k15_lattice3(A,B),u1_struct_0(A)) ) ),
    file(lattice3,k15_lattice3),
    []).

fof(dt_k1_group_1,axiom,(
    ! [A,B,C] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => m1_subset_1(k1_group_1(A,B,C),u1_struct_0(A)) ) ),
    file(group_1,k1_group_1),
    []).

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)) ) ),
    file(quantal1,k1_quantal1),
    []).

fof(dt_k1_xboole_0,axiom,(
    $true ),
    file(xboole_0,k1_xboole_0),
    []).

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)) ) ),
    file(quantal1,k2_quantal1),
    []).

fof(dt_k3_quantal1,axiom,(
    ! [A] : 
      ( l3_quantal1(A)
     => m1_subset_1(k3_quantal1(A),u1_struct_0(A)) ) ),
    file(quantal1,k3_quantal1),
    []).

fof(dt_k4_monoid_0,axiom,(
    ! [A,B,C] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => m1_subset_1(k4_monoid_0(A,B,C),u1_struct_0(A)) ) ),
    file(monoid_0,k4_monoid_0),
    []).

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)) ) ),
    file(quantal1,k5_quantal1),
    []).

fof(dt_l1_group_1,axiom,(
    ! [A] : 
      ( l1_group_1(A)
     => l1_struct_0(A) ) ),
    file(group_1,l1_group_1),
    []).

fof(dt_l1_lattices,axiom,(
    ! [A] : 
      ( l1_lattices(A)
     => l1_struct_0(A) ) ),
    file(lattices,l1_lattices),
    []).

fof(dt_l1_quantal1,axiom,(
    ! [A] : 
      ( l1_quantal1(A)
     => ( l3_lattices(A)
        & l1_group_1(A) ) ) ),
    file(quantal1,l1_quantal1),
    []).

fof(dt_l1_struct_0,axiom,(
    $true ),
    file(struct_0,l1_struct_0),
    []).

fof(dt_l1_vectsp_1,axiom,(
    ! [A] : 
      ( l1_vectsp_1(A)
     => l1_group_1(A) ) ),
    file(vectsp_1,l1_vectsp_1),
    []).

fof(dt_l2_lattices,axiom,(
    ! [A] : 
      ( l2_lattices(A)
     => l1_struct_0(A) ) ),
    file(lattices,l2_lattices),
    []).

fof(dt_l2_quantal1,axiom,(
    ! [A] : 
      ( l2_quantal1(A)
     => ( l1_quantal1(A)
        & l1_vectsp_1(A) ) ) ),
    file(quantal1,l2_quantal1),
    []).

fof(dt_l3_lattices,axiom,(
    ! [A] : 
      ( l3_lattices(A)
     => ( l1_lattices(A)
        & l2_lattices(A) ) ) ),
    file(lattices,l3_lattices),
    []).

fof(dt_l3_quantal1,axiom,(
    ! [A] : 
      ( l3_quantal1(A)
     => l2_quantal1(A) ) ),
    file(quantal1,l3_quantal1),
    []).

fof(dt_m1_subset_1,axiom,(
    $true ),
    file(subset_1,m1_subset_1),
    []).

fof(dt_u1_quantal1,axiom,(
    ! [A] : 
      ( l3_quantal1(A)
     => m1_subset_1(u1_quantal1(A),u1_struct_0(A)) ) ),
    file(quantal1,u1_quantal1),
    []).

fof(dt_u1_struct_0,axiom,(
    $true ),
    file(struct_0,u1_struct_0),
    []).

fof(existence_l1_group_1,axiom,(
    ? [A] : l1_group_1(A) ),
    file(group_1,l1_group_1),
    []).

fof(existence_l1_lattices,axiom,(
    ? [A] : l1_lattices(A) ),
    file(lattices,l1_lattices),
    []).

fof(existence_l1_quantal1,axiom,(
    ? [A] : l1_quantal1(A) ),
    file(quantal1,l1_quantal1),
    []).

fof(existence_l1_struct_0,axiom,(
    ? [A] : l1_struct_0(A) ),
    file(struct_0,l1_struct_0),
    []).

fof(existence_l1_vectsp_1,axiom,(
    ? [A] : l1_vectsp_1(A) ),
    file(vectsp_1,l1_vectsp_1),
    []).

fof(existence_l2_lattices,axiom,(
    ? [A] : l2_lattices(A) ),
    file(lattices,l2_lattices),
    []).

fof(existence_l2_quantal1,axiom,(
    ? [A] : l2_quantal1(A) ),
    file(quantal1,l2_quantal1),
    []).

fof(existence_l3_lattices,axiom,(
    ? [A] : l3_lattices(A) ),
    file(lattices,l3_lattices),
    []).

fof(existence_l3_quantal1,axiom,(
    ? [A] : l3_quantal1(A) ),
    file(quantal1,l3_quantal1),
    []).

fof(existence_m1_subset_1,axiom,(
    ! [A] : 
    ? [B] : m1_subset_1(B,A) ),
    file(subset_1,m1_subset_1),
    []).

fof(fc1_struct_0,axiom,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ~ v1_xboole_0(u1_struct_0(A)) ) ),
    file(struct_0,fc1_struct_0),
    []).

fof(fc1_xboole_0,axiom,(
    v1_xboole_0(k1_xboole_0) ),
    file(xboole_0,fc1_xboole_0),
    []).

fof(fraenkel_a_3_10_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_10_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) ) ) ) ),
    file(quantal1,a_3_10_quantal1),
    []).

fof(fraenkel_a_3_11_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_11_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) ) ) ) ),
    file(quantal1,a_3_11_quantal1),
    []).

fof(rc11_quantal1,axiom,(
    ? [A] : 
      ( l3_quantal1(A)
      & ~ v3_struct_0(A) ) ),
    file(quantal1,rc11_quantal1),
    []).

fof(rc1_finset_1,axiom,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_finset_1(A) ) ),
    file(finset_1,rc1_finset_1),
    []).

fof(rc1_xboole_0,axiom,(
    ? [A] : v1_xboole_0(A) ),
    file(xboole_0,rc1_xboole_0),
    []).

fof(rc2_quantal1,axiom,(
    ? [A] : 
      ( l1_quantal1(A)
      & ~ v3_struct_0(A) ) ),
    file(quantal1,rc2_quantal1),
    []).

fof(rc2_xboole_0,axiom,(
    ? [A] : ~ v1_xboole_0(A) ),
    file(xboole_0,rc2_xboole_0),
    []).

fof(rc3_struct_0,axiom,(
    ? [A] : 
      ( l1_struct_0(A)
      & ~ v3_struct_0(A) ) ),
    file(struct_0,rc3_struct_0),
    []).

fof(rc4_quantal1,axiom,(
    ? [A] : 
      ( l2_quantal1(A)
      & ~ v3_struct_0(A) ) ),
    file(quantal1,rc4_quantal1),
    []).

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) ) ),
    file(quantal1,rc7_quantal1),
    []).

fof(redefinition_k4_monoid_0,axiom,(
    ! [A,B,C] : 
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => k4_monoid_0(A,B,C) = k1_group_1(A,B,C) ) ),
    file(monoid_0,k4_monoid_0),
    []).

fof(t1_subset,axiom,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => m1_subset_1(A,B) ) ),
    file(subset,t1_subset),
    []).

fof(t2_subset,axiom,(
    ! [A,B] : 
      ( m1_subset_1(A,B)
     => ( v1_xboole_0(B)
        | r2_hidden(A,B) ) ) ),
    file(subset,t2_subset),
    []).

fof(t2_tarski,axiom,(
    ! [A,B] : 
      ( ! [C] : 
          ( r2_hidden(C,A)
        <=> r2_hidden(C,B) )
     => A = B ) ),
    file(tarski,t2_tarski),
    []).

fof(t6_boole,axiom,(
    ! [A] : 
      ( v1_xboole_0(A)
     => A = k1_xboole_0 ) ),
    file(boole,t6_boole),
    []).

fof(t7_boole,axiom,(
    ! [A,B] : ~ ( r2_hidden(A,B)
      & v1_xboole_0(B) ) ),
    file(boole,t7_boole),
    []).

fof(t8_boole,axiom,(
    ! [A,B] : ~ ( v1_xboole_0(A)
      & A != B
      & v1_xboole_0(B) ) ),
    file(boole,t8_boole),
    []).