If an even number is subtracted from an odd number, then the remainder is odd.

Let the even number *BC* be subtracted from the odd number *AB.*

I say that the remainder *CA* is odd.

Subtract the unit *AD,* therefore *DB* is even.

But *BC* is also even, therefore the remainder *CD* is even. Therefore *CA* is odd.

Therefore, *if an even number is subtracted from an odd number, then the remainder is odd.*

Q.E.D.