Difference between 2 Consecutive Cubes is Odd/Proof 2

Proof
Let $a, b \in \Z$ such that $b = a + 1$.

Either:
 * $a$ is even and $b$ is odd

or:
 * $b$ is even and $a$ is odd.

Hence from Parity of Integer equals Parity of Positive Power either:
 * $a^3$ is even and $b^3$ is odd

or:
 * $b^3$ is even and $a^3$ is odd.

The result follows.