theorem Th118:
  for S being set for X,Y being ManySortedSet of S st X c= Y
  holds Union coprod X c= Union coprod Y
proof
  let S be set;
  let X,Y be ManySortedSet of S such that
A1: X c= Y;
A2: dom Y = S by PARTFUN1:def 2;
  let x be object;
  assume
A3: x in Union coprod X;
  then
A4: x`2 in dom X by CARD_3:22;
A5: x`1 in X.x`2 by A3,CARD_3:22;
A6: x = [x`1,x`2] by A3,CARD_3:22;
  X.x`2 c= Y.x`2 by A1,A4;
  hence thesis by A2,A4,A5,A6,CARD_3:22;
end;
