If from an odd number an even number be subtracted, the remainder will be odd.
For from the odd number AB let the even number BC be subtracted;
I say that the remainder CA is odd.
Let the unit AD be subtracted;
therefore DB is even. [VII. Def. 7]
But BC is also even;
therefore the remainder CD is even. [ IX. 24 ]
Therefore CA is odd. [VII. Def. 7]
Q. E. D.