I have to prove the Lemma stated in the Title. What I have so far is the following:
Since two edges that form a cycle of length 2 are parallel edges, I need to prove that G\e contains such a cycle.
Let u and v be the two vertices incident with the parallel edges e and f. Then, let the degree of vertex of u and v each be 2. By 2-connectivity of G, the graph G\e is connected. Hence, there exists a path from u to v in G\e and this path together with edges e and f form the cycle containing the two vertices u and v that are incident with parallel edges e and f.
When I read through the lecture notes, I also found the following: If G/g has a loop, h say, then the edges g and h are parallel in G, in which case G\g is 2-connected by Lemma 3.3. Do I need to include that in my proof and if yes then how?
This prove seems so obvious that it makes it even harder to prove somehow lol
Thanks for the help!