In texts which present the third isomorphism theorem: $$(G/N)/(H/N) \cong G/H$$
the relationship between the entities is often seen presented in the form:
Let $H$ and $N$ be normal subgroups of a group $G$, and $N \subseteq H$. Then $H/N$ is a normal subgroup of $G/N$ and there is an isomorphism $(G/N)/(H/N) \cong G/H$.
(EDIT: This is how it is presented in Allan Clark's Elements of Abstract Algebra (1971), article $68$, for example.)
I have been in discussion with someone who insists that it is mandatory to state that $N$ is a normal subgroup of $H$, or the definition is invalid, because if $N$ is not a normal subgroup of $H$ then $H/N$ is undefined.
My own personal preference is that the statement of the theorem is perfectly good as it stands, because if $N$ and $H$ are both normal in $G$ and $N \subseteq H$, then it logically follows that $N$ is a normal subgroup of $H$ and it is therefore unnecessary and inelegant to specifically state this in the exposition.
What one would of course do is to derive the property during the course of the proof, or reference it as a pre-proven lemma.
Indeed, we have, by definition of normal subgroup: $$\forall g \in G: g H = H g$$
$$\forall g \in G: g N = N g$$
$$\forall g \in H: g N = N g$$
But, it is argued, the above is non-trivial and far from obvious, and therefore it is important in the exposition to make that explicit statement: "Let $N$ be a normal subgroup of $H$."
To my mind it is better to merely state subsetness, and to prove normal-subgroupness during the course of the proof itself. Otherwise it may leave the reader open to ask the question: "But what happens if $N$ is a subset of $H$ but not necessarily a normal subgroup of $H$? What happens then?" without taking that mental step of realisation that you can't have $N$ being a subset of $H$ under the above circumstances without $N$ being a normal subgroup of $H$.
In other words, I believe it important to establish that (on the surface) "more general" statement of the problem, and it is unhelpful to explicitly include one of the facts that is established during the course of the proof.
What is the general consensus of opinion here?