reserve I for set;

theorem
  for f being ManySortedFunction of I for X being ManySortedSet of I
  holds f.:.:X c= rngs f
proof
  let f be ManySortedFunction of I;
  let X be ManySortedSet of I;
  let i be object;
  assume
A1: i in I;
  then (f.:.:X).i = (f.i).:(X.i) by PBOOLE:def 20;
  then (f.:.:X).i c= rng (f.i) by RELAT_1:111;
  hence thesis by A1,MSSUBFAM:13;
end;
