Normal distribution with mean coming from normal distribution

Question

this is my first time writing here. I have been looking the whole day in this page and others and I couldn't find the answer, even if it seems trivial to me.

Given a normal distribution $N(m,s_1^2)$, what is the distribution of $N(N(m,s_1),s_2^2)$?

I know by simulation that it will follow $N(m,s_1^2+s_2^2)$. This property has a name? Where can I find a demonstration for this?

Thank you in advance.

By the way, I found this other post in which is explained: Density of a Normal RV whose mean is drawn from a Normal Distribution (Compound Distribution)

The explanation is the following: You will find that if $X∼N(μ,σ^2)$ and $Y∼N(X,τ^2)$ then $Y∼N(μ,σ^2+τ^2)$. Consider $Z=Y−X∼N(0,τ^2)$ independent of X, and then consider Y=X+Z. However I find unclear the step $Z=Y−X∼N(0,τ^2)$

Answer 1

Suppose $X = x$, fixed. Then $Y \sim N(x,\tau^2)$, so $Y - x \sim N(0,\tau^2)$. Basically what is said is that $Z\mid X = x \sim N(0,\tau^2)$ for every possible value of $x$, hence $Z \sim N(0,\tau^2)$, i.e. conditioning doesn't change the distribution.