Partial Derivative/Examples/u^2 + v^2 = x^2, 2 u v = 2 x y + y^2/Lemma

Lemma for Example: $u^2 + v^2 = x^2$, $2 u v = 2 x y + y^2$
Consider the simultaneous equations:

Then:


 * $x = 1$, $y = -2$ is a solution at $u = 1$, $v = 0$.

Proof
We make sure that $\tuple {1, -2}$ is actually a solution at $u = 1$, $v = 0$:

and it is seen that this is the case.