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:22
  g is monotonic & h is monotonic implies g * h is monotonic
proof
  assume that
A1: g is monotonic and
A2: h is monotonic;
A3: dom h = Bool M by FUNCT_2:def 1;
  let X, Y be Element of Bool M;
  assume X c= Y;
  then h.X c= h.Y by A2;
  then g.(h.X) c= g.(h.Y) by A1;
  then g.(h.X) c= (g*h).Y by A3,FUNCT_1:13;
  hence thesis by A3,FUNCT_1:13;
end;
