Use of prime symbol in proof writing Questions:
So I looked through my course notes and saw this proof. I understand the content, but I'm confused about the use of the prime symbol. If we say that there is some $j'$ such that it is a natural number, and $n = 2j' + 1$, what is the purpose of declaring $j$ on the following line?
Here is the proof of if n is odd, n$^2$ is also odd, from my notes:


*

*Assume $n \in \mathbb{N}$.

*

*Assume n is odd. 

*

*Then, $\exists j' \in \mathbb{N}, n = 2j' + 1$. # by definition of n
odd

*

*Let $j \in \mathbb{N}$ be such that $n = 2j + 1$. # name it j

*Then $n^2 = 4j^2 + 4j + 1 = 2(2j^2 + 2j) + 1$. # definition of $n^2$

*Then $\exists k \in \mathbb{N}; n^2 = 2k + 1.$ # $2j^2 + 2j \in \mathbb{N}$, since N closed under +;

*Then $n^2$ is odd. # by denition of $n^2$ odd



*Then n is odd $\implies$ n$^2$ is odd. 


*Then $\forall n \in \mathbb{N}, $n odd$ \implies n^2$ odd. 


A few more questions:
If done this way, could I simply just continue using $j'$ rather than declaring $j$? As in, could I just write $n^2 = 4j'^2 + 4j' + 1$ and skipped the let statement instead?
Is there a reason that $k$ is used instead of something like $j''$?
Thanks for your time. Apologies for the formatting.
 A: Statements $\exists a, \Phi(a)$ and $\exists b, \Phi(b)$ are equivalent and interchangeable (apart from technical details that the variables may not occur "hidden" in $\Phi$; for example $\exists a,a+3b=5$ is not equivalent to $\exists b,b+3b=5$). The same holds for $\forall a,\Phi(a)$ and $\forall b,\Phi(b)$.
Now regarding the rulkes of inference used: If we know $\exists a,\Phi(a)$, we are allowed to pick and work with an object, $b$ say, that makes this true, i.e. a $b$ such that $\Phi(b)$. It may be clearer to name this specific $b$ different from the $a$ used inthe quantifier. Then again, as we might have just as well started from $\exists b,\Phi(b)$ it seems that nothing forbids us to use the same variable name. The author chose to not keep the variable name and in order to end up with the desired $j$, he started with something different, namely $j'$.
Likewise at the end: If we derive $\Phi(x)$ for some expression $x$, then we may conclude $\exists a,\Phi(a)$ (with similar caveats about the variable $a$) and again the specific quantifier variable used doesn't matter. It should of course not be the same as some variable already defined in the current "scope" as one would say in programming. So $j''$ instead of $k$ would indeed be fine, but would that look nicer?
