
theorem Th24:
  for n being set, L being add-associative right_zeroed
right_complementable non empty addLoopStr, p being Series of n, L holds p-p =
  0_(n,L)
proof
  let n be set, L be add-associative right_zeroed right_complementable non
  empty addLoopStr, p be Series of n, L;
  reconsider pp = p-p, Z = 0_(n,L) as Function of Bags n, the carrier of L;
  now
    let b be Element of Bags n;
    thus pp.b = p.b+(-p).b by Th15
      .= p.b + -p.b by Th17
      .= 0.L by RLVECT_1:def 10
      .= Z.b;
  end;
  hence thesis;
end;
