Equivalence of Definitions of Odd-Times Odd Integer

$(1)$ implies $(2)$
Let $n$ be an odd-times odd integer by definition 1.

Then by definition:
 * $n > 1$
 * $n$ is not prime.

Thus:
 * $n = r s$

where $1 < r < p$ and $1 < s < p$.

As $n$ is odd:
 * $2 \nmid n$

and so:
 * $2 \nmid r$

and:
 * $2 \nmid s$

So both $r$ and $s$ are odd integers greater than $1$ such that $n = r s$.

Thus $n$ is an odd-times odd integer by definition 2.

$(2)$ implies $(1)$
Let $n$ be an odd-times odd integer by definition 2.

Then by definition:
 * $n = r s$

where $r > 1, s > 1$ are odd integers.

Thus $n$ is not prime by definition.

As $r$ and $s$ are odd:
 * $2 \nmid r$

and
 * $2 \nmid s$

and so:
 * $2 \nmid r s$

So $n = r s$ is odd.

As $r > 1$ and $s > 1$ it follows from Divisor Relation on Positive Integers is Partial Ordering that $n > 1$.

Thus $n$ is an odd-times odd integer by definition 1.