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

theorem Th38:
  for Amp being AmpleSet of R for a,b,c being Element of R holds
  c = gcd(a,b,Amp) & c <> 0.R implies gcd((a/c),(b/c),Amp) = 1.R
proof
  let Amp be AmpleSet of R;
  let A,B,C be Element of R;
  assume that
A1: C = gcd(A,B,Amp) and
A2: C <> 0.R;
  set A1 = A/C;
  C divides A by A1,Def12;
  then
A3: A1 * C = A by A2,Def4;
  set B1 = B/C;
A4: gcd((A1 * C),(B1 * C),Amp) is_associated_to (C * gcd(A1,B1,Amp)) by Th36;
A5: gcd(A1,B1,Amp) is Element of Amp & 1.R is Element of Amp by Def8,Def12;
  C divides B by A1,Def12;
  then gcd(A,B,Amp) = gcd((A1 * C),(B1 * C),Amp) by A2,A3,Def4;
  then (C * 1.R) is_associated_to (C * gcd(A1,B1,Amp)) by A1,A4;
  hence thesis by A2,A5,Th19,Th22;
end;
