I would like to get some help with my answer to this problem.
My answer:
First, consider if $a=b$. This automatically gives us that the statement is true.
Secondly, we consider when $a \neq b$.
Pick any $x \in A$.
Note that $x-b \in A-b$.
Also, $x-a \in A-a$, but since $b \in A$, $b-a \in A-a$.
Then, $x-a+a-b \in A-a$ by the fact that $A-a$ is a linear subspace.
Thus, we get $x-b \in A-a$.
This shows that $A-b \subseteq A-a$.
And, we get $A-a \subseteq A-b$ with similar reasoning.
Hence, $A-a=A-b$.
Although it looks correct to me, I am still doubtful with this answer because we normally say $A-b \subseteq A-a$ when we can say "if $x-b \in A-b$, then $x-b \in A-a$".
How should I reorder my answer to make it clearer? Or, if there's any error, how can it be fixed?