Klein-bottle and Möbius-strip together with a homeomorphism Consider the Klein bottle (this can be done by making a quotient space). I want to give a proof of the following statement:
The Klein Bottle is homeomorphic to the union of two copies of a Möbius strip joined by a homeomorphism along their boundaries.
I know what such a Möbius band looks like and how we can obtain this also by a quotient map. I also know how to see the Klein bottle, but I don't understand that the given statement is correct. How do you construct such a homeomorphism explicitly?

Present a Klein bottle as a square with vertical edges identified in an orientation-reversing manner and horizontal edges identified in an orientation-preserving manner. Now make two horizontal cuts at one-third of the way up and two-thirds of the way up. 
The middle third is one Möbius strip. Take the upper and lower thirds and glue them by the original horizontal identification. This is the other Möbius strip.
