If from an even number an odd number be subtracted, the remainder will be odd.

For from the even number AB let the odd number BC be subtracted; I say that the remainder CA is odd.

For let the unit CD be subtracted from BC; therefore DB is even. [VII. Def. 7]

But AB is also even; therefore the remainder AD is also even. [IX. 24]

And CD is an unit; therefore CA is odd. [VII. Def. 7] Q. E. D.