reserve S for non empty non void ManySortedSign;
reserve X for non-empty ManySortedSet of S;

theorem Th1:
  for I being set, f1,f2 being ManySortedSet of I st f1 c= f2
  holds Union f1 c= Union f2
  proof
    let I be set;
    let f1,f2 be ManySortedSet of I;
    assume A1: f1 c= f2;
    let x be object;
    assume x in Union f1; then
    consider y being object such that
A2: y in dom f1 & x in f1.y by CARD_5:2;
A3: dom f1 = I & dom f2 = I by PARTFUN1:def 2;
    f1.y c= f2.y by A1,A2;
    hence x in Union f2 by A2,A3,CARD_5:2;
  end;
