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.