Currently working through Apostol need a check on my logic

VectorField

1. The problem statement, all variables and given/known data
Prove the following theorem.
If a≠0, then (a^-1)^-1=a

Theorems proven before/axioms I am allowed to use:
Existence of reciprocals axiom: there exist real numbers x and y where x≠0 such that xy=1
Possibility of division Thm: basically a^-1=1/a




2. Relevant equations
I just need a check on my proof since I am self studying the book. Any logical mistakes on my proof or suggestions etc? This is my first time writing proofs.


3. The attempt at a solution
Assume (a^-1)^-1=a is true.
Then by the existence of reciprocals axiom there exists an x such that (a^-1)^-1x=1.
Hence,
Now by the possibility of division theorem we have,
Finally since we assumed (a^-1)^-1=a was true it follows that:

It looks fine to me, I'm just a noob at proofs. Thanks.

SammyS

According to the first line of your proof,

Assume (a^-1)^-1=a is true.

you're assuming that the thing you're proving is true.

That's a definite No-No .

I suggest starting with:

a^-1 has a reciprocal, (a^-1)^-1.

Therefore, (a^-1)^-1 a^-1 = 1
...

Now, multiply that by a, with a being on the right.