I am trying to prove the following:
Let $G$ be a $2-$connected graph but not a triangle, and let $e$ be an edge of $G$. Show that either $G − e$ or $G/e$ is again $2-$connected.
I found an answer here. In general I don't have problems with the proof (I think it lacks a case when we suppose $x,y$ are in $P_1$ but $xy$ is not, but even in that case it can be easily solved removing the cycle which forms, however I may be mistaken and that case might not be possible for other reason).
The only problem I find is when they say
Otherwise (If $G-e$ is not connected) $x$ and $y$ must be connected by only one path $P_{xy}$ in $G − e$.
I don't know why is that. Couldn't there be two non-independent paths between $x$ and $y$? I think I may be missing some step here. My question is what is it that I'm missing here?