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

theorem Th42:
  for Amp,Amp9 being AmpleSet of R for x,y being Element of R st
    x,y are_canonical_wrt Amp holds x,y are_canonical_wrt Amp9
proof
  let Amp,Amp9 be AmpleSet of R;
  let x,y be Element of R;
  1.R * x = x;
  then
A1: 1.R divides x;
  1.R * y = y;
  then
A2: 1.R divides y;
  assume x,y are_canonical_wrt Amp;
  then gcd(x,y,Amp) = 1.R;
  then
A3: for z being Element of R st z divides x & z divides y holds z divides 1.
  R by Def12;
  1.R in Amp9 by Def8;
  then gcd(x,y,Amp9) = 1.R by A3,A1,A2,Def12;
  hence thesis;
end;
