reserve x, y for object, I for set,
  A, B, X, Y for ManySortedSet of I;

theorem :: FIN_TOPO:4
  X c= A iff X in bool A
proof
  thus X c= A implies X in bool A
  proof
    assume
A1: X c= A;
    let i be object;
    assume
A2: i in I;
    then X.i c= A.i by A1;
    then X.i in bool (A.i);
    hence thesis by A2,Def1;
  end;
  assume
A3: X in bool A;
  let i be object;
  assume
A4: i in I;
  then X.i in (bool A).i by A3;
  then X.i in bool (A.i) by A4,Def1;
  hence thesis;
end;
