reserve A for set, x,y,z for object,
  k for Element of NAT;
reserve n for Nat,
  x for object;
reserve V, C for set;

theorem Th55:
  for n being set, b1, b2 be bag of n st b1 divides b2 & b2 -' b1
  = EmptyBag n holds b2 = b1
proof
  let n be set, b1, b2 be bag of n such that
A1: b1 divides b2 and
A2: b2 -' b1 = EmptyBag n;
  now
    let k be object;
    assume k in n;
    0 = (b2-'b1).k by A2
      .= b2.k -' b1.k by Def6;
    then
A3: b2.k <= b1.k by NAT_D:36;
    b1.k <= b2.k by A1;
    hence b2.k = b1.k by A3,XXREAL_0:1;
  end;
  hence thesis;
end;
