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 monotonic & R is monotonic implies P ** R is monotonic
proof
  assume that
A1: P is monotonic and
A2: R is monotonic;
A3: doms R = bool M by MSSUBFAM:17;
  let X, Y be Element of bool M;
  assume X c= Y;
  then R..X c= R..Y by A2;
  then P..(R..X) c= P..(R..Y) by A1;
  then P..(R..X) c= (P**R)..Y by A3,Th4,MSSUBFAM:12;
  hence thesis by A3,Th4,MSSUBFAM:12;
end;
