
theorem Th4:
  for n being set, L being add-associative right_zeroed
right_complementable non empty addLoopStr, p be Series of n, L holds p - 0_(n
  ,L) = p
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-0_(n,L) as Function of Bags n,the carrier of L;
  now
    let b be Element of Bags n;
    thus pp.b = (p+-0_(n,L)).b by POLYNOM1:def 7
      .= p.b + (-0_(n,L)).b by POLYNOM1:15
      .= p.b + -((0_(n,L)).b) by POLYNOM1:17
      .= p.b + -0.L by POLYNOM1:22
      .= p.b - 0.L
      .= p.b by RLVECT_1:13;
  end;
  hence thesis by FUNCT_2:63;
end;
