
theorem Th1:
  for X being set, L being Abelian add-associative right_zeroed
right_complementable non empty addLoopStr, p,q being Series of X, L holds -(p
  + q) = (-p) + (-q)
proof
  let n be set, L be Abelian add-associative right_zeroed right_complementable
  non empty addLoopStr, p,q be Series of n,L;
A1: now
    let x be object;
    assume x in dom -(p+q);
    then reconsider b = x as bag of n;
    ((-p) + (-q)).b = (-p).b + (-q).b by POLYNOM1:15
      .= -(p.b) + (-q).b by POLYNOM1:17
      .= -(p.b) + -(q.b) by POLYNOM1:17
      .= -(q.b + p.b) by RLVECT_1:31
      .= -((p+q).b) by POLYNOM1:15
      .= (-(p+q)).b by POLYNOM1:17;
    hence (-(p+q)).x = ((-p) + (-q)).x;
  end;
  dom (-(p+q)) = Bags n by FUNCT_2:def 1
    .= dom ((-p) + (-q)) by FUNCT_2:def 1;
  hence thesis by A1,FUNCT_1:2;
end;
