Two unequal magnitudes being set out, if from the greater there is subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process is repeated continually, then there will be left some magnitude less than the lesser magnitude set out.

Let *AB* and *C* be two unequal magnitudes of which *AB* is the greater.

I say that, if from *AB* there is subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process is repeated continually, then there will be left some magnitude which is less than the magnitude *C*.

cf. V.Def.4

Some multiple *DE* of *C* is greater than *AB*.

Divide *DE* into the parts *DF*, *FG*, and *GE* equal to *C*. From *AB* subtract *BH* greater than its half, and from *AH* subtract *HK* greater than its half, and repeat this process continually until the divisions in *AB* are equal in multitude with the divisions in *DE*.

Let, then, *AK*, *KH*, and *HB* be divisions equal in multitude with *DF*, *FG*, and *GE*.

Now, since *DE* is greater than *AB*, and from *DE* there has been subtracted *EG* less than its half, and, from *AB*, *BH* greater than its half, therefore the remainder *GD* is greater than the remainder *HA*.

And, since *GD* is greater than *HA*, and there has been subtracted from *GD* the half *GF*, and from *HA*, *HK* greater than its half, therefore the remainder *DF* is greater than the remainder *AK*.

But *DF* equals *C*, therefore *C* is also greater than *AK*. Therefore *AK* is less than *C*.

Therefore there is left of the magnitude *AB* the magnitude *AK* which is less than the lesser magnitude set out, namely *C*.

Therefore, *two unequal magnitudes being set out, if from the greater there is subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process is repeated continually, then there will be left some magnitude less than the lesser magnitude set out.*

Q.E.D.

And the theorem can similarly be proven even if the parts subtracted are halves.