>> Why not? It seems at this point
    >> you are assuming something that is
    >> generally deduced as a consequence
    >> of the FTA.

    > According to the OP, this result
    > does not rely on the FTA -- he
    > claims to derive it from the
    > Euclidean Algorithm.

Yes.

    > I can't do that, so I'm taking
    > his word for it, but that doesn't
    > make the proof circular.

But you should say that you are relying on this. As it is you are simply making an unsupported assertion, and so your proof is incomplete.

See other comments in this sub-thread for more explanations.