
theorem div1:
for F being Field,
    p,q being Polynomial of F st deg p < deg q holds p mod q = p
proof
let F be Field, p,q be Polynomial of F;
assume A1: deg p < deg q;
0_.(F) = 0.Polynom-Ring(F) by POLYNOM3:def 10; then
H: -(0_.(F)) = - 0.Polynom-Ring(F) by lm .= 0.Polynom-Ring(F);
A0: now assume A0: q = 0_.(F);
    then deg p < - 1 by A1,HURWITZ:20;
    hence contradiction by A0,A1,T8b;
    end;
p = 0_.(F) + p by POLYNOM3:28 .= ((0_.(F)) *' q) + p by POLYNOM3:34;
hence p mod q = p - (0_.(F)) *' q by A0,A1,HURWITZ:def 5
             .= p - 0_.(F) by POLYNOM3:34
             .= p + 0_.(F) by H,POLYNOM3:def 10
             .= p by POLYNOM3:28;
end;
