reserve I for set;

theorem
  for A, B being ManySortedSet of I st A is_transformable_to B for f
  being ManySortedFunction of I st doms f = A & rngs f c= B holds
  f is ManySortedFunction of A, B
proof
  let A, B be ManySortedSet of I such that
 for i being set st i in I holds B.i = {} implies A.i = {};
  let f be ManySortedFunction of I such that
A1: doms f = A and
A2: rngs f c= B;
  let i be object;
  assume
A3: i in I;
  then reconsider J = I as non empty set;
  reconsider s = i as Element of J by A3;
A4: dom (f.s) = A.s by A1,MSSUBFAM:14;
  rng (f.s) = (rngs f).s by MSSUBFAM:13;
  then rng (f.s) c= B.s by A2;
  hence thesis by A4,FUNCT_2:2;
end;
