
theorem
for F being Field
for p being irreducible Element of the carrier of Polynom-Ring F
for q being Element of the carrier of Polynom-Ring F
st q divides p holds q is unital or q is_associated_to p
proof
let F be Field;
let p be irreducible Element of the carrier of Polynom-Ring F;
let q be Element of the carrier of Polynom-Ring F;
assume AS: q divides p; then
consider r being Polynomial of F such that  C: q *' r = p by RING_4:1;
per cases;
suppose q is zero;
  hence thesis by C;
  end;
suppose D: q is non zero;
per cases by AS,RING_4:41;
suppose deg q < 1; then
  deg q + 1 <= 1 by INT_1:7; then
  (deg q + 1) - 1 <= 1 - 1 by XREAL_1:9; then
  q is constant by RING_4:def 4;
  hence thesis by D;
  end;
suppose A: deg q >= deg p;
  deg q <= deg p by AS,RING_5:13; then
  B: deg q = deg p by A,XXREAL_0:1;
  r <> 0_.(F) by C; then
  deg p = deg p + deg r by D,C,B,HURWITZ:23; then
  reconsider r as constant Element of the carrier of Polynom-Ring F
     by RING_4:def 4,POLYNOM3:def 10;
  F: r is non zero by C;
  q * r = p by C,POLYNOM3:def 10;
  hence thesis by F,GCD_1:18;
  end;
end;
end;
