| If an even number is subtracted from an even number, then the remainder is even. | ||
| Let the even number BC be subtracted from the even number AB.
I say that the remainder CA is even. |
||
| Since AB is even, therefore it has a half part. For the same reason BC also has a half part, so that the remainder CA also has a half part, and CA is therefore even. | VII.Def.6 | |
| Therefore, if an even number is subtracted from an even number, then the remainder is even. | ||
| Q.E.D. | ||
Book IX Introduction - Proposition IX.23 - Proposition IX.25.