The theorem states:
Any Pythagorean plane is isomorphic to the Cartesian plane $F^2$ over its field $F$ of segments.
Can anyone give me a reference for this theorem?
I am surmising that your definition for a Pythagorean plane is the same as mine, i.e. the one in Greenberg's excellent article.
That is precisely what Hilbert carried out in his book Foundations of geometry.
This book is free to the public online at the Project Gutenberg link above. It is not the most modern prose, but it's readable and at a good level of rigor.
I think I've heard some people dissatisfied with some details in Hilbert's book, so if you find that's the case, check in Hartshorne's Euclid and beyond. It's a great read and I believe it also produces the proof you want.