reserve i, x, I for set,
  A, M for ManySortedSet of I,
  f for Function,
  F for ManySortedFunction of I;

theorem
  id (bool A) is topological MSSetOp of A
proof
  reconsider a = id (bool A) as MSSetOp of A;
  a is topological
  proof
    let X, Y be Element of bool A;
    Y in bool A by MSSUBFAM:12;
    then
A1: Y c= A by MBOOLEAN:1;
    X in bool A by MSSUBFAM:12;
    then X c= A by MBOOLEAN:1;
    then X (\/) Y c= A by A1,PBOOLE:16;
    then X (\/) Y in bool A by MBOOLEAN:1;
    then X (\/) Y is Element of bool A by MSSUBFAM:11;
    hence thesis by Th10;
  end;
  hence thesis;
end;
