For starters make n+1 odd as follows
2(n+1)+1 now show the previous number is even.
T.P. n is even iff n+1 is odd
Assume n is even then let n = 2k for some k element of Z (definition of even)
Then n+1 = 2k + 1, but 2k + 1 is the definition of an odd number. This proves the forward direction now reverse it.
Let n + 1 be odd then by definition of an odd number n + 1 = 2k + 1 for an arbitrary k element of Z
Now subtract 1 from both sides and you have n = 2k which by definition is even.
This completes the proof in both directions.