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

theorem
  for I being non empty set
  for J being set
  for A,B being ManySortedSet of I st A c= B
  for f being Function of J,I holds A*f c= B*f
  proof
    let I be non empty set;
    let J be set;
    let A,B be ManySortedSet of I;
    assume A1: A c= B;
    let f be Function of J,I;
    let x be object; assume A2: x in J; then
    reconsider i = f.x as Element of I by FUNCT_2:5;
    (A*f).x = A.i & (B*f).x = B.i by A2,FUNCT_2:15;
    hence (A*f).x c= (B*f).x by A1;
  end;
