IX.25

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

Proof

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.