Sphere with three Möbius strips glued and sphere with a handle and a Möbius strip glued

Question

I am reading the first chapter from Topology by Armstrong. There, after stating the classification theorem for closed surfaces, he has mentioned an example that a sphere with one handle and one Möbius strip glued is homeomorphic to a sphere with three Möbius strips glued.

I am not able to see this. I know that to prove the above is to say that a torus with a Möbius strip glued is homeomorphic to a Klein bottle with a Möbius strip glued. How do I prove this? Or how do I at least convince myself that this is the case, in case I don't yet have the tools to prove it.

Answer 1

See the proof of Lemma 7.1 in Massey's "A Basic Course in Algebraic Topology" (1991), pp 23-25.