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 Th47:
  for X being set, b1, b2 being bag of X holds b2 + b1 -' b1 = b2
proof
  let X be set, b1, b2 be bag of X;
  now
    let k be object;
    assume k in X;
    thus (b2 + b1 -' b1).k = (b2+b1).k -' b1.k by Def6
      .= b2.k+b1.k -' b1.k by Def5
      .= b2.k by NAT_D:34;
  end;
  hence thesis;
end;
