reserve i, x, I for set,
  A, B, M for ManySortedSet of I,
  f, f1 for Function;
reserve SF, SG for SubsetFamily of M;
reserve E, T for Element of Bool M;

theorem :: CLOSURE:15
  id (Bool A) is topological SetOp of A
proof
  reconsider f = id (Bool A) as SetOp of A;
  f is topological
  proof
    let X, Y be Element of Bool A;
    X c= A & Y c= A by PBOOLE:def 18;
    then X (\/) Y c= A by PBOOLE:16;
    then X (\/) Y is ManySortedSubset of A by PBOOLE:def 18;
    then X (\/) Y in Bool A by Def1;
    hence f.(X (\/) Y) = X (\/) Y by FUNCT_1:18
      .= f.X (\/) Y
      .= f.X (\/) f.Y;
  end;
  hence thesis;
end;
