In the previous section we have seen that the GCD of polynomials in "R"["X"] may be deduced from GCDs in "R" and in "F"["X"]. A closer look on the proof shows that this allows us to prove the existence of GCDs in "R"["X"], if they exist in "R" and in "F"["X"]. In particular, if GCDs exist in "R", and if "X" is reduced to one variable, this proves that GCDs exist in "R"["X"] (Euclid's algorithm proves the existence of GCDs in "F"["X"]).