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

theorem Th32:
  for Amp being AmpleSet of R for a being Element of R holds
    gcd(a,1.R,Amp) = 1.R & gcd(1.R,a,Amp) = 1.R
proof
  let Amp be AmpleSet of R;
  let A be Element of R;
  1.R * A = A;
  then
A1: 1.R divides A;
  1.R in Amp & for z being Element of R st z divides A & z divides 1.R
  holds z divides 1.R by Def8;
  then gcd(A,1.R,Amp) = 1.R by A1,Def12;
  hence thesis by Th29;
end;
