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 Th22:
  for I being set
  for A,B,C being non-empty ManySortedSet of I
  for f being ManySortedFunction of A,B st B is ManySortedSubset of C
  holds f is ManySortedFunction of A,C
  proof
    let I be set;
    let A,B,C be non-empty ManySortedSet of I;
    let f be ManySortedFunction of A,B;
    assume A1: B c= C;
    let x be object; assume
A2: x in I; then
A3: f.x is Function of A.x,B.x & B.x <> {} & B.x c= C.x
    by A1;
    dom(f.x) = A.x & rng(f.x) c= B.x
    by A2,FUNCT_2:def 1;
    hence f.x is Function of A.x,C.x by A3,XBOOLE_1:1,FUNCT_2:2;
  end;
