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

theorem Th34:
  for Amp being AmpleSet of R for a,b,c being Element of R holds
  b is_associated_to c implies
    gcd(a,b,Amp) is_associated_to gcd(a,c,Amp) &
    gcd(b,a,Amp) is_associated_to gcd(c,a,Amp)
proof
  let Amp be AmpleSet of R;
  let A,B,C be Element of R;
A1: gcd(A,B,Amp) divides B by Def12;
A2: gcd(A,B,Amp) divides A by Def12;
A3: gcd(A,C,Amp) divides A by Def12;
A4: gcd(A,C,Amp) divides C by Def12;
A5: gcd(A,B,Amp) = gcd(B,A,Amp) by Th29;
  assume
A6: B is_associated_to C;
  then C divides B;
  then gcd(A,C,Amp) divides B by A4,Th2;
  then
A7: gcd(A,C,Amp) divides gcd(A,B,Amp) by A3,Def12;
  B divides C by A6;
  then gcd(A,B,Amp) divides C by A1,Th2;
  then gcd(A,B,Amp) divides gcd(A,C,Amp) by A2,Def12;
  then gcd(A,B,Amp) is_associated_to gcd(A,C,Amp) by A7;
  hence thesis by A5,Th29;
end;
