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

theorem Th33:
  for Amp being AmpleSet of R for a,b being Element of R holds
    gcd(a,b,Amp) = 0.R iff a = 0.R & b = 0.R
proof
  let Amp be AmpleSet of R;
  let A,B be Element of R;
A1: now
    assume
A2: gcd(A,B,Amp) = 0.R;
    then 0.R divides B by Def12;
    then
A3: ex E being Element of R st 0.R * E = B;
    0.R divides A by A2,Def12;
    then ex D being Element of R st 0.R * D = A;
    hence gcd(A,B,Amp) = 0.R implies A = 0.R & B = 0.R by A3;
  end;
  A = 0.R & B = 0.R implies gcd(A,B,Amp) = 0.R
  proof
    assume that
A4: A = 0.R and
A5: B = 0.R;
    gcd(A,B,Amp) = NF(A,Amp) by A5,Th30;
    hence thesis by A4,Th25;
  end;
  hence thesis by A1;
end;
