reserve i,j,k,n,m for Nat,
        X for set,
        b,s for bag of X,
        x for object;

theorem
   for b be bag of X holds 0 (#) b = EmptyBag X
proof
  let b be bag of X;set E=EmptyBag X;
  A1:dom (0 (#) b) = X = dom E & X=dom b by PARTFUN1:def 2;
  for x st x in dom E holds E.x=(0 (#) b).x
  proof
    let x;
    (0 (#)b).x = 0* b.x by VALUED_1:6;
    hence thesis;
  end;
  hence thesis by A1;
end;
