reserve S for non empty non void ManySortedSign;
reserve X for non-empty ManySortedSet of S;
reserve x,y,z for set, i,j for Nat;

theorem Th19:
  for S being set
  for X being ManySortedSet of S
  for Y being non-empty ManySortedSet of S
  for w being ManySortedFunction of X, Y
  holds rngs w is ManySortedSubset of Y
  proof
    let S be set;
    let X be ManySortedSet of S;
    let Y be non-empty ManySortedSet of S;
    let w be ManySortedFunction of X, Y;
    let x be object; assume x in S; then
    (rngs w).x = rng(w.x) by MSSUBFAM:13;
    hence (rngs w).x c= Y.x;
  end;
