reserve x,y,X,X1,Y,Z for set;
reserve L for Lattice;
reserve F,H for Filter of L;
reserve p,q,r for Element of L;
reserve L1, L2 for Lattice;
reserve a1,b1 for Element of L1;
reserve a2 for Element of L2;
reserve f for Homomorphism of L1,L2;
reserve B for Element of Fin the carrier of L;
reserve DL for distributive Lattice;
reserve f for Homomorphism of DL,L2;
reserve 0L for lower-bounded Lattice,
  B,B1,B2 for Element of Fin the carrier of 0L,
  b for Element of 0L;
reserve f for UnOp of the carrier of 0L;
reserve 1L for upper-bounded Lattice,
  B,B1,B2 for Element of Fin the carrier of 1L,
  b for Element of 1L;
reserve f,g for UnOp of the carrier of 1L;
reserve DL for distributive upper-bounded Lattice,
  B for Element of Fin the carrier of DL,
  p for Element of DL,
  f for UnOp of the carrier of DL;
reserve CL for C_Lattice;
reserve IL for implicative Lattice;
reserve f for Homomorphism of IL,CL;
reserve i,j,k for Element of IL;
reserve BL for Boolean Lattice;
reserve f for Homomorphism of BL,CL;
reserve A for non empty Subset of BL;
reserve a1,a,b,c,p,q for Element of BL;
reserve B,B0,B1,B2,A1,A2 for Element of Fin the carrier of BL;
reserve F,H for Field of BL;

theorem
  for AF being non empty ClosedSubset of BL st
  Bottom BL in AF & Top BL in AF holds
    { FinMeet B : B c= SetImp AF } = field_by AF
proof
  let AF be non empty ClosedSubset of BL such that
A1: Bottom BL in AF and
A2: Top BL in AF;
  set A1= { FinMeet B :B c= SetImp AF };
A3: AF c= A1
  proof
    let x be object;
    assume
A4: x in AF;
    then reconsider b=x as Element of BL;
    reconsider B = {.b.} as Element of Fin the carrier of BL;
    b = Bottom BL "\/" b
      .= (Top BL)` "\/" b by Th29;
    then b in SetImp AF by A2,A4,Th37;
    then
A5: B c= SetImp AF by ZFMISC_1:31;
    x = FinMeet B by Th10;
    hence thesis by A5;
  end;
  A1 c= the carrier of BL
  proof
    let x be object;
    assume x in A1;
    then ex B st x = FinMeet B & B c= SetImp AF;
    hence thesis;
  end;
  then reconsider A1 as non empty Subset of BL by A3;
A6: now
    let F;
    assume
A7: AF c= F;
    thus A1 c= F
    proof
      reconsider F1=F as ClosedSubset of BL by Th35;
      let x be object;
      assume x in A1;
      then consider B such that
A8:   x=FinMeet B and
A9:   B c= SetImp AF;
      SetImp AF c= F
      proof
        let y be object;
        assume y in SetImp AF;
        then ex p,q st y = p => q & p in AF & q in AF;
        hence thesis by A7,Th33;
      end;
      then B c= F1 by A9;
      hence thesis by A2,A7,A8,Th24;
    end;
  end;
  A1 is Field of BL
  proof
    let p,q;
    assume that
A10: p in A1 and
A11: q in A1;
    thus p "/\" q in A1
    proof
      consider B2 such that
A12:  q=FinMeet B2 & B2 c= SetImp AF by A11;
      consider B1 such that
A13:  p=FinMeet B1 & B1 c= SetImp AF by A10;
      consider B0 such that
A14:  B0=B1 \/ B2;
      B0 c= SetImp AF & p "/\" q = FinMeet B0 by A13,A12,A14,Th23,XBOOLE_1:8;
      hence thesis;
    end;
    thus p` in A1
    proof
      consider B such that
A15:  p=FinMeet B and
A16:  B c= SetImp AF by A10;
      p` = FinJoin ( B,comp BL) by A15,Th41;
      then ex B0 st B0 c= SetImp AF & p` = FinMeet B0 by A1,A2,A16,Th42;
      hence thesis;
    end;
  end;
  hence thesis by A3,A6,Def10;
end;
