reserve i, x, I for set,
  A, M for ManySortedSet of I,
  f for Function,
  F for ManySortedFunction of I;
reserve P, R for MSSetOp of M,
  E, T for Element of bool M;

theorem
  P is topological & R is topological implies P ** R is topological
proof
  assume that
A1: P is topological and
A2: R is topological;
  let X, Y be Element of bool M;
A3: doms R = bool M by MSSUBFAM:17;
  Y in bool M by MSSUBFAM:12;
  then
A4: Y c= M by MBOOLEAN:1;
  X in bool M by MSSUBFAM:12;
  then X c= M by MBOOLEAN:1;
  then X (\/) Y c= M by A4,PBOOLE:16;
  then X (\/) Y in doms R by A3,MBOOLEAN:1;
  hence (P**R)..(X (\/) Y) = P..(R..(X (\/) Y)) by Th4
    .= P..(R..X (\/) R..Y) by A2
    .= P..(R..X) (\/) P..(R..Y) by A1
    .= (P**R)..X (\/) P..(R..Y) by A3,Th4,MSSUBFAM:12
    .= (P**R)..X (\/) (P**R)..Y by A3,Th4,MSSUBFAM:12;
end;
