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.