Despite labels $a$, $b$, $c$, I don't see how it's fully general. At best you could say it works for similar triangles, if you have that concept.

But I don't see how it proves anything if $a$ were say doubled, and $b$ kept the same.

I'm not sure what you mean, sorry. It's definitely fully general. It demonstrates that:

* For any real positive values $a$ and $b$

* let $T$ be the (unique) right-angled triangle with non-hypotenuse sides of length $a$ and $b$

* then $a$^2 = $b$^2 = $c$^2

* where $c$ is the (unique) length of the hypotenuse of $T$.

You could give me the instance

  $a$ = 2, $b$ = 3

and it will prove it for this case. Then you could later double $a$ and give me the instance

  $a$ = 4, $b$ = 3

and it will prove it just as easily for this case.