
theorem DZIW:
for R being Ring
for p being Polynomial of R
for q being Element of the carrier of Polynom-Ring R
st p = q holds -p = -q
proof
let R be Ring; let p be Polynomial of R;
let q be Element of the carrier of Polynom-Ring R;
assume AS: p = q;
reconsider r = -p as Element of the carrier of Polynom-Ring R
  by POLYNOM3:def 10;
now let n be Element of NAT;
  thus (p + -p).n = p.n + (-p).n by NORMSP_1:def 2
                 .= p.n + -(p.n) by BHSP_1:44
                 .= (0_.(R)).n by RLVECT_1:5;
  end;
then p + -p = 0_.(R);
then q + r = 0_.(R) by AS,POLYNOM3:def 10;
then q + r = 0.(Polynom-Ring R) by POLYNOM3:def 10;
hence thesis by RLVECT_1:6;
end;
