
theorem Th3:
  for X being set, L being add-associative right_zeroed
right_complementable non empty addLoopStr, p being Series of X, L holds -p +
  p = 0_(X,L) & p + -p = 0_(X,L)
proof
  let n be set, L be add-associative right_zeroed right_complementable non
  empty addLoopStr, p be Series of n, L;
  set q = -p + p;
  now
    let b be Element of Bags n;
    thus q.b = (-p).b + p.b by POLYNOM1:15
      .= -(p.b) + p.b by POLYNOM1:17
      .= 0.L by RLVECT_1:5
      .= (0_(n,L)).b by POLYNOM1:22;
  end;
  hence -p + p = 0_(n,L) by FUNCT_2:63;
  set q = p + -p;
  now
    let b be Element of Bags n;
    thus q.b = p.b + (-p).b by POLYNOM1:15
      .= p.b + -p.b by POLYNOM1:17
      .= 0.L by RLVECT_1:5
      .= (0_(n,L)).b by POLYNOM1:22;
  end;
  hence thesis by FUNCT_2:63;
end;
