Odd Number multiplied by Odd Number is Odd

Theorem

 * If an odd number by multiplying an odd number make some number, the product will be odd.

Proof
Let $a$ be odd and $b$ be even.

Then by definition of odd number:
 * $\exists c \in \Z: a = 2 c + 1$
 * $\exists d \in \Z: b = 2 d + 1$

So:

Hence the result by definition of odd number.