Let D, E, F be numbers called by the same name as the parts A, B, C, and let G, the least number measured by D, E, F, be taken. [[VII. 36](elem.7.36)]

Therefore G has parts called by the same name as D, E, F. [[VII. 37](elem.7.37)]

But A, B, C are parts called by the same name as D, E, F; therefore G has the parts A, B, C.

I say next that it is also the least number that has.

For, if not, there will be some number less than G which will have the parts A, B, C.

Let it be H.

Since H has the parts A, B, C, therefore H will be measured by numbers called by the same name as the parts A, B, C. [[VII. 38](elem.7.38)]

But D, E, F are numbers called by the same name as the parts A, B, C; therefore H is measured by D, E, F.

And it is less than G : which is impossible.

Therefore there will be no number less than G that will have the parts A, B, C. Q. E. D.