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

theorem Th28:
  for Amp being AmpleSet of R for a,b,c being Element of R holds
  c divides gcd(a,b,Amp) implies c divides a & c divides b
proof
  let Amp be AmpleSet of R;
  let A,B,C be Element of R;
  assume C divides gcd(A,B,Amp);
  then consider D being Element of R such that
A1: C * D = gcd(A,B,Amp);
  gcd(A,B,Amp) divides A by Def12;
  then consider E being Element of R such that
A2: gcd(A,B,Amp) * E = A;
A3: C * (D * E) = A by A1,A2,GROUP_1:def 3;
  gcd(A,B,Amp) divides B by Def12;
  then consider E being Element of R such that
A4: gcd(A,B,Amp) * E = B;
  C * (D * E) = B by A1,A4,GROUP_1:def 3;
  hence thesis by A3;
end;
