reserve X,Y for set;
reserve R for domRing-like commutative Ring;
reserve c for Element of R;
reserve R for gcdDomain;

theorem Th30:
  for Amp being AmpleSet of R for a being Element of R holds
    gcd(a,0.R,Amp) = NF(a,Amp) & gcd(0.R,a,Amp) = NF(a,Amp)
proof
  let Amp be AmpleSet of R;
  let A be Element of R;
A1: NF(A,Amp) in Amp by Def9;
  NF(A,Amp) * 0.R = 0.R;
  then
A2: NF(A,Amp) divides 0.R;
A3: NF(A,Amp)is_associated_to A by Def9;
A4: for z being Element of R st z divides A & z divides 0.R holds z divides
  NF(A,Amp)
  by A3,Th2;
  NF(A,Amp) divides A by A3;
  then gcd(A,0.R,Amp) = NF(A,Amp) by A2,A4,A1,Def12;
  hence thesis by Th29;
end;
