If a rational area is subtracted from a medial area, then there arise two other irrational straight lines, either a first apotome of a medial straight line or a straight line which produces with a rational area a medial whole.

Let the rational area *BD* be subtracted from the medial area *BC*.

I say that the side of the remainder *EC* becomes one of two irrational straight lines, either a first apotome of a medial straight line or a straight line which produces with a rational area a medial whole.

Set out a rational straight line *FG*, and apply the areas similarly. Then *FH* is rational and incommensurable in length with *FG*, while *KF* is rational and commensurable in length with *FG*, therefore *FH* and *FK* are rational straight lines commensurable in square only.

Therefore *KH* is an apotome, and *FK* the annex to it.

Now the square on *HF* is greater than the square on *FK* either by the square on a straight line commensurable with *HF* or by the square on a straight line incommensurable with it.

If the square on *HF* is greater than the square on *FK* by the square on a straight line commensurable with *HF*, while the annex *FK* is commensurable in length with the rational straight line *FG* set out, then *KH* is a second apotome.

But *FG* is rational, so that the side of *LH*, that is, of *EC*, is a first apotome of a medial straight line.

But, if the square on *HF* is greater than the square on *FK* by the square on a straight line incommensurable with *HF*, while the annex *FK* is commensurable in length with the rational straight line *FG* set out, then *KH* is a fifth apotome, so that the side of *EC* is a straight line which produces with a rational area a medial whole.

Therefore, *if a rational area is subtracted from a medial area, then there arise two other irrational straight lines, either a first apotome of a medial straight line or a straight line which produces with a rational area a medial whole.*

Q.E.D.