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

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

For, since AB is odd, let the unit BD be subtracted; therefore the remainder AD is even. [VII. Def. 7]

For the same reason CD is also even; [VII. Def. 7] so that the remainder CA is also even. [IX. 24] Q. E. D.