I have noticed that many authors tend to use "let" instead of "for all". For example, they write something like this:

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> Let $n$ be an even natural number. Then also $n^2$ is even.
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I wonder, why they use "let" instead of "for all", also in cases where the "for all"-version sounds quite good:

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> For all even natural numbers $n$, $n^2$ is even.
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> 
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Note that "let" has a slightly different meaning than "for all":

The statement "Let $n$ be an even natural number. Then also $n^2$ is even" translated into the logic calculus would be something like:
$\mathrm{even}(n)\vdash \mathrm{even}(n\cdot n)$ (this means that "$\mathrm{even}(n\cdot n)$" is true when we are supposing that "$\mathrm{even}(n)$" holds). On the other hand, the statement "For all even natural numbers $n$, $n^2$ ist even" can be translated into a single formula $\forall n.\ \mathrm{even}(n)\implies \mathrm{even}(n\cdot n)$.
EDIT: In the formalization of the examples the quantifier $\forall$ ranges only over natural numbers, so this is the type of "object" we are considering.

I think that in most cases the second version ("For all ...") is meant, but the authors however use "let".

Here is my question:

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> Why do so many authors write their statements in the form "Let [Variable] be a [Type]. Then ...", even when they actually mean "for all" and even when the version written "for all ..." sounds quite good?
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Here a example where this causes confusion:

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> Theorem: Let $G$ be a planar graph, and let $V$ be the number of vertices, $E$ the number of edges and $F$ the number of faces. Then $V-E+F = 2$.
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Proof: by induction on the number of edges $E$.

Why is this confusing? Because a proof by induction gives us a "for all"-statement.

Maybe I take the formalization of proofs too serious and exact. In this case: Sorry for the question.