Suppose G(V,E) is a multigraph(i.e. we can have parallel edges in the graph) which contains a cycle C and for all vertices $v \in V$ we have that $deg(v)$ is even.
Now suppose that we remove C from G to obtain G'(V',E').
Why is it the cases that all of the remaining vertices still have even degree?
I understand that if v was not in C then it's degree is unchanged but what if v was adjacent(with one edge) to a vertex u in C? Surely then v would be left with an odd degree?