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Proof of Biconditional Involving Even and Odd Integers.

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.

http://img241.imageshack.us/img241/9840/310p1dp2.jpg

http://img96.imageshack.us/img96/753/310p2qt9.jpg

http://img514.imageshack.us/img514/2442/310p3ht7.jpg

http://img170.imageshack.us/img170/5516/310p4mi1.jpg