TSTST 2014 Problem 1

Let $\leftarrow$ denote the left arrow key on a standard keyboard. If one opens a text editor and types the keys "ab $\leftarrow \mathrm{cd} \leftarrow \leftarrow \mathrm{e} \leftarrow \leftarrow \mathrm{f}$ ", the result is "faecdb". We say that a string $B$ is reachable from a string $A$ if it is possible to insert some amount of $\leftarrow$ 's in $A$, such that typing the resulting characters produces $B$. So, our example shows that "faecdb" is reachable from "abcdef". Prove that for any two strings $A$ and $B, A$ is reachable from $B$ if and only if $B$ is reachable from $A$.