
theorem
for R being non degenerated Ring,
    p being non zero Polynomial of R
for q being Polynomial of R holds deg(p*'q) <= deg p + deg q
proof
let R be non degenerated Ring;
let p be non zero Polynomial of R; let q be Polynomial of R;
   deg p = len p - 1 & deg q = len q - 1 &
   deg(p*'q) = len(p*'q) - 1 by HURWITZ:def 2;
then C: deg(p*'q) + 1 <= (deg p + 1) + (deg q + 1) -' 1 by leng;
reconsider degp = deg p as Element of NAT;
D: deg p + 1 >= 0 + 1 by XREAL_1:6;
deg q + 1 >= 0
  proof
  per cases;
  suppose q = 0_.(R);
    then deg q = -1 by HURWITZ:20;
    hence thesis;
    end;
  suppose q <> 0_.(R);
    then reconsider q1 = q as non zero Polynomial of R by UPROOTS:def 5;
    reconsider degq = deg q1 as Element of NAT;
    degq >= 0;
    hence thesis;
    end;
  end;
then (deg p + 1) + (deg q + 1) >= 1 + 0 by D,XREAL_1:7;
then (deg p + 1) + (deg q + 1) - 1 >= 1 - 1 by XREAL_1:9;
then deg(p*'q) + 1 <= (deg p + 1) + deg q by C,XREAL_0:def 2;
then deg(p*'q) + 1  - 1 <= (deg p + 1) + deg q - 1 by XREAL_1:9;
hence thesis;
end;
