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

Let $n$ be an even natural number. Then also $n^2$ is even.

I wonder, why they use "let" instead of "for all", also in cases where the "for all"-version sounds quite good:

For all even natural numbers $n$, $n^2$ is even.

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:

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?

Here a example where this causes confusion:

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 $V-E+F = 2$.

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.
 A: The authors do that for two reasons first, to give the reader a breather, second, because they want do do more with the notation than just finish this one sentence.
After "Let $n$ ne a natrual number" there is a pause. A pause in which time the notation sinks in, so that people transfer it from their ultrashort memory to their short memory so that it can then be used for various purposes. In particular, the notation then has a longer half-life than the notation in "For every even natural number $n$, the number $n^2$ ist even."
In this last sentence, the meaning of $n$ being a natural number, is erased with the period. Not so in the previous case, where it can be used on.
A: The "Let ... Then ... " statement is an abuse of language which is also grammatically incorrect. See http://www.math.illinois.edu/~dwest/grammar.html#letthen.
A: Language, written and spoken, is a flexible beast. English is particularly flexible. There are often many grammatically correct ways to say the same thing. Good, careful writing requires the ability to use this flexibility and to rein it in as appropriate, but this takes a lot of time and requires lots and lots of self-editing. Quick, not-so-careful writing exploits this flexibility, depending on the flexibility of the language to avoid loss of information. Both the careful and the quick ways of writing are useful and important in professional mathematical writing.
For example, consider your two sentences:

Let $n$ be an even natural number. Then also $n^2$ is even.
For all even natural numbers $n$, $n^2$ ist even.

You, and I, and others in this thread, and probably most other experienced mathematical readers, understand the meanings of these two sentences, and we probably all get the same mathematical information from reading them. We probably even instantaneously spell-check "ist" and get "is".
Now it may be that some logical parser tranlates these two sentences into different statements of some symbolic calculus.
But, your parser and my brain might be different. And I, writing the first sentence, might be trying to convey something different than I, writing the second sentence. I might have some didactic reason for writing it one way rather than the other, despite the more "efficient" or "correct" or "machine readable" advantage the other has over the one.
For example, I might have various reasons for expressing a universal quantifier in the fashion of "Let". If you will indulge me, here is one thing I might be trying to convey:

Let $n$ be an even natural number. Any one at all. Like, even one with a gazillion digits. I'm not just talking about $2$ or $4$ or $6$ here!!! No matter WHAT even natural number $n$ we take, also $n^2$ is even.

I'm not trying to be silly here, I'm just trying to point out that conveying a mathematical idea in a human fashion (as opposed to a machine fashion) sometimes requires different modes of expression.
