
theorem divmod:
for F being Field,
    p being Polynomial of F
for q being non zero Polynomial of F holds deg(p mod q) < deg q
proof
let F be Field, p be Polynomial of F; let q be non zero Polynomial of F;
q <> 0_.(F); then
consider t being Polynomial of F such that
C: p = (p div q) *' q + t & deg t < deg q by HURWITZ:def 5;
p mod q = ((p div q) *' q + t) - ((p div q) *' q) by C,HURWITZ:def 6
       .= t + ((p div q) *' q - ((p div q) *' q)) by POLYNOM3:26
       .= t + 0_.(F) by POLYNOM3:29;
hence thesis by C;
end;
