Question about the proof of simple algebra rule $\frac{1}{\frac{1}{a}} = a$ I have a question about a proof I saw in a book about basic algeba rules. The  rule to prove is: 
\begin{eqnarray*}
\frac{1}{\frac{1}{a}} = a, \quad a \in \mathbb{R}_{\ne 0}
\end{eqnarray*}
And the proof: 
\begin{eqnarray*}
1 = a \frac{1}{a} \Longrightarrow 1 =  \frac{1}{a} \frac{1}{\frac{1}{a}} \Longrightarrow a = a \frac{1}{a} \frac{1}{\frac{1}{a}} \Longrightarrow \frac{1}{\frac{1}{a}} = a
\end{eqnarray*}
Why is it allowed to just replace $a$ with $1/a$? What's the explanation behind it? 

Let $x=\frac{1}{a}$. Then:
\begin{eqnarray*}
1 = x \frac{1}{x} \Longrightarrow 1 =  \frac{1}{a} \frac{1}{\frac{1}{a}} \Longrightarrow a = a \frac{1}{a} \frac{1}{\frac{1}{a}} \Longrightarrow \frac{1}{\frac{1}{a}} = a
\end{eqnarray*}
Better? You're right, replacing $a\to\frac{1}{a}$ is a minor abuse of notation, because they're implicitly changing the variable without telling you. But you can fix that by just letting $x=\frac{1}{a}$ at the start.
