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;
reserve g, h for SetOp of M;

theorem :: CLOSURE:19
  for A being Element of Bool M st A = E (\) T holds g is topological
  implies g.E (\) g.T c= g.A
proof
  let A be Element of Bool M such that
A1: A = E (\) T;
  assume
A2: g is topological;
  then g.E (\/) g.T = g.(E (\/) T)
    .= g.((E(\)T) (\/) T) by PBOOLE:67
    .= (g.A) (\/) (g.T) by A1,A2;
  then g.E c= g.A (\/) g.T by PBOOLE:14;
  then g.E (\) g.T c= (g.A (\/) g.T) (\) g.T by PBOOLE:53;
  then
A3: g.E (\) g.T c= g.A (\) g.T by PBOOLE:75;
  g.A (\) g.T c= g.A by PBOOLE:56;
  hence thesis by A3,PBOOLE:13;
end;
