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

theorem Th43:
  for Amp being AmpleSet of R for x,y being Element of R holds
  x,y are_co-prime implies gcd(x,y,Amp) = 1.R
proof
  let Amp be AmpleSet of R;
  let x,y be Element of R;
  assume x,y are_co-prime;
  then consider Amp9 being AmpleSet of R such that
A1: gcd(x,y,Amp9) = 1.R;
  x,y are_canonical_wrt Amp9 by A1;
  then x,y are_canonical_wrt Amp by Th42;
  hence thesis;
end;
