
theorem T88a:
for R being domRing
for p being Element of the carrier of Polynom-Ring R
holds (ex q being Element of the carrier of Polynom-Ring R
                                 st q divides p & 1 <= deg q & deg q < deg p)
implies p is reducible
proof
let R be domRing, p be Element of the carrier of Polynom-Ring R;
set K = Polynom-Ring R;
reconsider pp = p as Polynomial of R;
assume ex q being Element of the carrier of Polynom-Ring R
                             st q divides p & 1 <= deg q & deg q < deg p;
then consider q being Element of the carrier of Polynom-Ring R such that
A0: q divides p & 1 <= deg q & deg q < deg p;
A1: not q is Unit of K by T88,A0;
now assume q is_associated_to p;
      then consider r being Element of K such that
      A2: r is unital & q * r = p by GCD_1:18;
      reconsider rr = r as Element of the carrier of K;
      A5:  p = q *' rr by A2,POLYNOM3:def 10;
      then A4: r <> 0_.(R) by HURWITZ:20,A0,POLYNOM3:34;
      q <> 0_.(R) by A5,A0,POLYNOM3:34;
      then deg p = deg q + deg rr by A4,A5,HURWITZ:23 .= deg q + 0 by A2,T88;
      hence contradiction by A0;
      end;
hence p is reducible by A0,A1,RING_2:def 9;
end;
