Proof: By Euclid
(related to Corollary: Cor. 10.111: Thirteen Irrational Straight Lines of Different Order)
 For the (square) on a medial (straight line), applied to a rational (straight line), produces as breadth a rational (straight line which is) incommensurable in length with the (straight line) to which (the area) is applied [Prop. 10.22].
 And the (square) on an apotome, applied to a rational (straight line), produces as breadth a first apotome [Prop. 10.97].
 And the (square) on a first apotome of a medial (straight line), applied to a rational (straight line), produces as breadth a second apotome [Prop. 10.98].
 And the (square) on a second apotome of a medial (straight line), applied to a rational (straight line), produces as breadth a third apotome [Prop. 10.99].
 And (square) on a minor (straight line), applied to a rational (straight line), produces as breadth a fourth apotome [Prop. 10.100].
 And (square) on that (straight line) which with a rational (area) produces a medial whole, applied to a rational (straight line), produces as breadth a fifth apotome [Prop. 10.101].
 And (square) on that (straight line) which with a medial (area) produces a medial whole, applied to a rational (straight line), produces as breadth a sixth apotome [Prop. 10.102].
 Therefore, since the aforementioned breadths differ from the first (breadth), and from one another  from the first, because it is rational, and from one another since they are not the same in order^{1}  clearly, the irrational (straight lines) themselves also differ from one another.
 And since it has been shown that an apotome is not the same as a binomial [Prop. 10.111], and (that) the (irrational straight lines) after the apotome, being applied to a rational (straight line), produce as breadth, each according to its own (order), apotomes, and (that) the (irrational straight lines) after the binomial themselves also (produce as breadth), according (to their) order, binomials, the (irrational straight lines) after the apotome are thus different, and the (irrational straight lines) after the binomial (are also) different, so that there are, in order, 13 irrational (straight lines) in all:
 Medial,
 Binomial,
 First bimedial,
 Second bimedial,
 Major,
 Square root of a rational plus a medial (area) ,
 Square root of (the sum of) two medial (areas),
 Apotome,
 First apotome of a medial,
 Second apotome of a medial,
 Minor,
 That which with a rational (area) produces a medial whole,
 That which with a medial (area) produces a medial whole.
∎
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Footnotes