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 reflexive & for X being Element of bool M holds P..X c= X)
  implies P is idempotent
proof
  assume that
A1: P is reflexive and
A2: for X being Element of bool M holds P..X c= X;
  let X be Element of bool M;
A3: P..X c= P..(P..X) by A1;
   P..(P..X) c= P..X by A2;
  hence thesis by A3,PBOOLE:146;
end;
