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

theorem Th37:
  for Amp being AmpleSet of R for a,b,c being Element of R holds
  gcd(a,b,Amp) = 1.R implies gcd(a,(b * c),Amp) = gcd(a,c,Amp)
proof
  let Amp be AmpleSet of R;
  let A,B,C be Element of R;
  assume gcd(A,B,Amp) = 1.R;
  then
A1: C * (gcd(A,B,Amp)) = C;
  gcd((A * C),(B * C),Amp) is_associated_to (C * gcd(A,B,Amp)) by Th36;
  then gcd(A,C,Amp) is_associated_to gcd(A,gcd((A * C),(B * C),Amp),Amp) by A1
,Th34;
  then
A2: gcd(A,C,Amp) is_associated_to gcd(gcd(A,(A * C),Amp),(B * C),Amp) by Th35;
A3: A * gcd(1.R,C,Amp) = A * 1.R by Th32
    .= A;
  gcd((A * 1.R),(A * C),Amp) is_associated_to (A * gcd(1.R,C,Amp)) by Th36;
  then gcd(A,(A * C),Amp) is_associated_to A by A3;
  then
A4: gcd(gcd(A,(A * C),Amp),(B * C),Amp) is_associated_to gcd(A,(B * C),Amp)
  by Th34;
  gcd(A,(B * C),Amp) is Element of Amp & gcd(A,C,Amp) is Element of Amp
  by Def12;
  hence thesis by A2,A4,Th4,Th22;
end;
