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 implies P..E (\) P..T c= P..(E (\) T)
proof
  E in bool M by MSSUBFAM:12;
  then E c= M by MBOOLEAN:1;
  then E(\)T c= M by MBOOLEAN:15;
  then E(\)T in bool M by MBOOLEAN:1;
  then
A1: E(\)T is Element of bool M by MSSUBFAM:11;
  assume
A2: P is topological;
  then P..E (\/) P..T = P..(E (\/) T)
    .= P..((E(\)T) (\/) T) by PBOOLE:67
    .= (P..(E(\)T)) (\/) (P..T) by A1,A2;
  then P..E c= P..(E(\)T) (\/) P..T by PBOOLE:14;
  then P..E (\) P..T c= (P..(E(\)T) (\/) P..T) (\) P..T by PBOOLE:53;
  then
A3: P..E (\) P..T c= P..(E(\)T) (\) P..T by PBOOLE:75;
  P..(E(\)T) (\) P..T c= P..(E(\)T) by PBOOLE:56;
  hence thesis by A3,PBOOLE:13;
end;
