
theorem lemX:
for R being Ring
for p1,p2 being Polynomial of R holds p1 *' (-p2) = -(p1 *' p2)
proof
let R be Ring, p1,p2 being Polynomial of R;
reconsider a = p1, b = p2 as Element of the carrier of Polynom-Ring R
   by POLYNOM3:def 10;
A: a * b = p1 *' p2 by POLYNOM3:def 10;
-p2 = -b by FIELD_8:1;
hence p1 *' (-p2) = a * (-b) by POLYNOM3:def 10
                 .= -(a * b) by VECTSP_1:8
                 .= -(p1 *' p2) by A,FIELD_8:1;
end;
