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 Th46:
  for n being set, b1, b2 being bag of n st b1 divides b2
  holds b2 -' b1 + b1 = b2
proof
  let n be set, b1, b2 be bag of n such that
A1: b1 divides b2;
  now
    let k be object;
    assume k in n;
A2: b1.k <= b2.k by A1;
    thus (b2 -' b1 + b1).k = (b2-'b1).k + b1.k by Def5
      .= b2.k -' b1.k + b1.k by Def6
      .= b2.k + b1.k -' b1.k by A2,NAT_D:38
      .= b2.k by NAT_D:34;
  end;
  hence thesis;
end;
