
theorem div4:
for F being Field,
    p,r being Polynomial of F
for q being non zero Polynomial of F
holds (p + r) div q = (p div q) + (r div q) &
      (p + r) mod q = (p mod q) + (r mod q)
proof
let F be Field, p,r be Polynomial of F;
let q be non zero Polynomial of F;
H: q <> 0_.(F);
A: p + r
   = ((p div q) *' q + (p mod q)) + r by div5
  .= ((p div q) *' q + (p mod q)) + ((r div q) *' q + (r mod q)) by div5
  .= (p div q) *' q + ((p mod q) + ((r div q) *' q + (r mod q)))
     by POLYNOM3:26
  .= (p div q) *' q + (((p mod q) + (r mod q)) + ((r div q) *' q))
     by POLYNOM3:26
  .= ((p div q) *' q + (r div q) *' q) + ((p mod q) + (r mod q))
     by POLYNOM3:26
  .= ((p div q) + (r div q)) *' q + ((p mod q) + (r mod q))
     by POLYNOM3:32;
B: deg((p mod q) + (r mod q)) <= max(deg(p mod q),deg(r mod q))
   by HURWITZ:22;
C: deg(p mod q) < deg q by RING_2:29;
deg(r mod q) < deg q by RING_2:29;
then max(deg(p mod q),deg(r mod q)) < max(deg q,deg q) by C,XXREAL_0:27;
then D: deg((p mod q) + (r mod q)) < deg q by B,XXREAL_0:2;
hence (p + r) div q = (p div q) + (r div q) by A,H,HURWITZ:def 5;
thus (p + r) mod q
  = ((p div q) + (r div q)) *' q + ((p mod q) + (r mod q)) -
    (((p div q) + (r div q)) *' q) by D,A,H,HURWITZ:def 5
 .= (((p div q) + (r div q)) *' q - (((p div q) + (r div q)) *' q)) +
    ((p mod q) + (r mod q)) by POLYNOM3:26
 .= 0_.(F) + ((p mod q) + (r mod q)) by POLYNOM3:29
 .= (p mod q) + (r mod q) by POLYNOM3:28;
end;
