
theorem degA:
for L being Field
for p being Polynomial of L
for q being non zero Polynomial of L holds deg(p mod q) < deg q
proof
let L be Field;
let p be Polynomial of L;
let q be non zero Polynomial of L;
set u = p div q;
q <> 0_.(L); then
consider r being Polynomial of L such that
A: p = u *' q + r & deg r < deg q by HURWITZ:def 5;
p mod q = p - u *'q by HURWITZ:def 6
       .= u *' q + r + -(u *'q) by A,POLYNOM3:def 6
       .= (u *' q + -(u*' q)) + r by POLYNOM3:26
       .= (u *' q - u*' q) + r by POLYNOM3:def 6
       .= 0_.(L) + r by POLYNOM3:29
       .= r by POLYNOM3:28;
hence thesis by A;
end;
