
theorem Th21:
  for n being set, L being add-associative right_zeroed non empty
  doubleLoopStr, p, q, r being Series of n, L holds (p+q)+r = p+(q+r)
proof
  let n be set, L be add-associative right_zeroed non empty doubleLoopStr, p
  , q, r be Series of n, L;
  now
    let b be Element of Bags n;
    thus ((p+q)+r).b = (p+q).b+r.b by Th15
      .= p.b+q.b+r.b by Th15
      .= p.b+(q.b+r.b) by RLVECT_1:def 3
      .= p.b+(q+r).b by Th15
      .= (p+(q+r)).b by Th15;
  end;
  hence thesis;
end;
