
theorem div2:
for F being Field,
    p,q being Polynomial of F holds p mod q = 0_.(F) iff q divides p
proof
let F be Field, p,q be Polynomial of F;
A: now assume p mod q = 0_.(F);
   then (p div q) *' q
          = (p - (p div q) *' q) + (p div q) *' q by POLYNOM3:28
         .= p + (-((p div q) *' q) + (p div q) *' q) by POLYNOM3:26
         .= p + ((p div q) *' q - (p div q) *' q)
         .= p + 0_.(F) by POLYNOM3:29
         .= p by POLYNOM3:28;
   hence q divides p by T2;
   end;
now assume A: q divides p;
  thus p mod q = p - p by A,div0
              .= 0_.(F) by POLYNOM3:29;
  end;
hence thesis by A;
end;
