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:25
  g is topological & h is topological implies g * h is topological
proof
  assume that
A1: g is topological and
A2: h is topological;
  let X, Y be Element of Bool M;
A3: dom h = Bool M by FUNCT_2:def 1;
  hence (g*h).(X (\/) Y) = g.(h.(X (\/) Y)) by Th10,FUNCT_1:13
    .= g.(h.X (\/) h.Y) by A2
    .= g.(h.X) (\/) g.(h.Y) by A1
    .= (g*h).X (\/) g.(h.Y) by A3,FUNCT_1:13
    .= (g*h).X (\/) (g*h).Y by A3,FUNCT_1:13;
end;
