I did not look into much details but it looks right.
What main result are your trying to prove?
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The only complaint I have is that you leave TOO much room. You just look at how I wrote my homework! If I can find a scanner I will show you.
Could anyone verify or critique this proof? Throughout the proof, I am using the fact that an even integer can be written in the form of x=2m where m is an integer, and an odd integer has the form y=2n+1 where n is an integer.