$\sum_{q=1}^\infty (\sum_{p=1}^q a_{pq}) = \sum_{p=1}^\infty (\sum_{q=p}^\infty a_{pq}) = \sum_{(p,q)\in M} a_{pq}$?

Question

I am reading "Lectures on Complex Function Theory" by Takaaki Nomura.

There are the following two propositions in this book:

Let $\sum_{(p,q)\in \mathbb{N}^2} a_{pq}$ be a double series of positive terms.

1. If $\sum_{(p,q)\in \mathbb{N}^2} a_{pq} = S \in \mathbb{R}$, then $\sum_{q=1}^\infty (\sum_{p=1}^\infty a_{pq}) = S$ and $\sum_{p=1}^\infty (\sum_{q=1}^\infty a_{pq}) = S$.
2. If $\sum_{q=1}^\infty (\sum_{p=1}^\infty a_{pq}) \in \mathbb{R}$ or $\sum_{p=1}^\infty (\sum_{q=1}^\infty a_{pq}) \in \mathbb{R}$, then $\sum_{q=1}^\infty (\sum_{p=1}^\infty a_{pq}) = \sum_{p=1}^\infty (\sum_{q=1}^\infty a_{pq}) = \sum_{(p,q)\in \mathbb{N}^2} a_{pq}$.

I wonder the following two similar propositions hold or not.

Let $M := \{(p, q) \in \mathbb{N}^2 \mid p \leq q\}$ and $\sum_{(p,q)\in M} a_{pq}$ be a double series of positive terms.

1. If $\sum_{(p,q)\in M} a_{pq} = S \in \mathbb{R}$, then $\sum_{q=1}^\infty (\sum_{p=1}^q a_{pq}) = S$ and $\sum_{p=1}^\infty (\sum_{q=p}^\infty a_{pq}) = S$.
2. If $\sum_{q=1}^\infty (\sum_{p=1}^q a_{pq}) \in \mathbb{R}$ or $\sum_{p=1}^\infty (\sum_{q=p}^\infty a_{pq}) \in \mathbb{R}$, then $\sum_{q=1}^\infty (\sum_{p=1}^q a_{pq}) = \sum_{p=1}^\infty (\sum_{q=p}^\infty a_{pq}) = \sum_{(p,q)\in M} a_{pq}$.