Construction of Rational Straight Lines Commensurable in Square Only whose Square Differences Commensurable with Greater/Lemma 1

Proof
Let $a$ and $b$ be two natural numbers such that $a > b$.

Let $a$ and $b$ be either both even or both odd.

From:
 * Even Number minus Even Number is Even

and:
 * Odd Number minus Odd Number is Even

their difference $c = a - b$ is even.

Let $d = \dfrac c 2$.

Let $a$ and $b$ be similar plane numbers.

By definition of similar plane numbers it is noted that $a$ and $b$ may both be square.

Then by Square of Sum less Square:
 * $a b + d^2 = \left({a - d}\right)^2$

From Product of Similar Plane Numbers is Square, $a b$ is a square number.

So two square numbers $a b$ and $d^2$ have been found whose sum is square.

Similarly:
 * $a b = \left({a - d}\right)^2 - d^2$

from which it is noted that two square numbers $\left({a - d}\right)^2$ and $d^2$ have been found whose difference is square.