reserve n for Nat;

theorem prl25:
for R being domRing,
    p being Polynomial of R
for q being non zero Polynomial of R st p divides q holds deg p <= deg q
proof
let R be domRing, p be Polynomial of R;
let q being non zero Polynomial of R;
assume p divides q;
then consider r being Polynomial of R such that
A: q = p *' r by RING_4:1;
C: p <> 0_.(R) & r <> 0_.(R) by A;
then reconsider dq = deg q, dr = deg r, dp = deg p as Element of NAT by T8;
dq = dr + dp by A,C,HURWITZ:23;
hence thesis by NAT_1:11;
end;
