
theorem np00:
for R being domRing,
    p,q being Element of the carrier of Polynom-Ring R
holds p is_associated_to q implies deg p = deg q
proof
let R be domRing,
    p,q be Element of the carrier of Polynom-Ring R;
assume A: p is_associated_to q;
then consider c being Element of Polynom-Ring R such that
K2: q = p * c by GCD_1:def 1;
reconsider r = c as Element of the carrier of Polynom-Ring R;
consider d being Element of Polynom-Ring R such that
K3: p = q * d by A,GCD_1:def 1;
reconsider s = d as Element of the carrier of Polynom-Ring R;
K4: q = p *' r by K2,POLYNOM3:def 10;
K5: p = q *' s by K3,POLYNOM3:def 10;
per cases;
suppose p = 0_.(R);
  hence thesis by K4,POLYNOM3:34;
  end;
suppose q = 0_.(R);
  hence thesis by K5,POLYNOM3:34;
  end;
suppose A: p <> 0_.(R) & q <> 0_.(R);
  then A1: r <> 0_.(R) & s <> 0_.(R) by K4,K5,POLYNOM3:34;
  then A2: deg r is Element of NAT & deg s is Element of NAT by T8b;
  A3: deg q = deg p + deg r & deg p = deg q + deg s
           by A,A1,K4,K5,HURWITZ:23;
  then deg r + deg s = 0;
  then deg r = 0 & deg s = 0 by A2;
  hence thesis by A3;
  end;
end;
