
theorem lem22:
for F being Field
for p being non constant Element of the carrier of Polynom-Ring F
ex q being non constant monic Element of the carrier of Polynom-Ring F
st q divides p & q is irreducible
proof
let F be Field, p be non constant Element of the carrier of Polynom-Ring F;
defpred Q[Nat] means
  for p be non constant Element of the carrier of Polynom-Ring F
  st deg p = $1
  ex q being non constant monic Element of the carrier of Polynom-Ring F
  st q divides p & q is irreducible;
IS: now let k be Nat;
    assume AS1: for n being Nat st n < k holds Q[n];
      now let p be non constant Element of the carrier of Polynom-Ring F;
        assume AS3: deg p = k;
        per cases;
        suppose AS4: p is irreducible;
          set q = NormPolynomial p;
          p *' 1_.(F) = p; then
          A: q divides p by RING_4:25,RING_4:1;
             q is irreducible by AS4,RING_4:28;
          hence ex q being non constant monic
                              Element of the carrier of Polynom-Ring F
                st q divides p & q is irreducible by A;
          end;
        suppose AS4: p is non irreducible;
          p <> 0_.(F); then
          consider q being Element of the carrier of Polynom-Ring F such that
          A: q divides p & 1 <= deg q & deg q < deg p by AS4,RING_4:41;
          reconsider m = deg q as Element of NAT by A,INT_1:3;
          H: Q[m] by AS1,A,AS3;
          q is non constant by A,RING_4:def 4; then
          consider r being non constant monic
                           Element of the carrier of Polynom-Ring F such that
          B: r divides q & r is irreducible by H;
          consider s being Polynomial of F such that
          C: q *' s = p by A,RING_4:1;
          consider t being Polynomial of F such that
          D: r *' t = q by B,RING_4:1;
          r *' (s *' t) = p by C,D,POLYNOM3:33;
          hence ex q being non constant monic
                              Element of the carrier of Polynom-Ring F
              st q divides p & q is irreducible by B,RING_4:1;
          end;
        end;
      hence Q[k];
      end;
I: for k being Nat holds Q[k] from NAT_1:sch 4(IS);
reconsider n = deg p as Element of NAT;
Q[n] by I;
hence thesis;
end;
