A short reply:
Last night I think I realized where the sticking point is in many of the comments recieved to date: Is it valid in Case 3 to use Case 1? I say it is valid because Case 1 is the only congruent number relation, i.e., 1 = 1, of the three relations of the Trichotomy Law. And since the assumption of equality in Case 3 requires a congruent number relation, it requires the use of Case 1. Otherwise, we would be working with non-congruent numbers, i.e., 1 < > 1, which would be meaningless.
The purpose of, ± u^2, in Cases 2 and 3 is to ensure congruent number units.
Yes, but the problem is that case 1 applies ONLY IF [itex]x^2 + y^2 = z^2[/itex] and in case 3 you assume that [itex]x^2 + y^2 > z^2[/itex]. So, I don't know where case 1 comes from. Also, I am confuesd about why you expend so much energy in case 3 to deduce that [itex]x^2 + y^2 > z^2[/itex] when this was assumed at the start. Why can't you apply case 1 right then and there?
Also, I am concerned that it took you roughly 20 lines for case 1 and case 2 - which are trivial - and about 5 or so lines for case 3 - which is the hard case. This right there sends up red flags.