
theorem div3:
for F being Field,
    r,q,u being Polynomial of F
for p being non zero Polynomial of F
holds (((r *' q) mod p) *' u) mod p = (r *' q *' u) mod p
proof
let F be Field, r,q,u be Polynomial of F;
let p be non zero Polynomial of F;
A: (((r *' q) mod p) *' u) mod p
   = ((r *' q) *' u + (-((r *' q) div p) *' p) *' u) mod p by POLYNOM3:32
  .= (((r *' q) *' u) mod p) + (((-((r *' q) div p) *' p) *' u) mod p)
     by div4;
(-((r *' q) div p) *' p) *' u
   = (p *' (-((r *' q) div p))) *' u by HURWITZ:12
  .= p *' ((-((r *' q) div p)) *' u) by POLYNOM3:33;
hence (((r *' q) mod p) *' u) mod p
    = ((r *' q) *' u) mod p + 0_.(F) by A,div2,T2
   .= (r *' q *' u) mod p by POLYNOM3:28;
end;
