An equivalent response to yours would be: Your response is meaningless because you haven’t defined what an integer is.
After reproducing my proof in a very wordy and obtuse manner which no instructor would use, you ask:
"Why must an integer be either even or odd? Why can't it be BOTH, or nether?",
Fair enough. By the Euclidean algorithm any integer on division by 2 must leave a remainder of 0 or1. If 0, even; if 1, odd, by definition.
ie, I=2m or I=2n+1, which I thought would be accepted at a high school level. I see no point in belaboring it.
I am still confused as to why this endless rehashing of my original proof, except for the question above, which I am glad to answer.