Proof: By Euclid
(related to Lemma: Lem. 11.23: Making a Square Area Equal to the Difference Of Areas of Two Other Incongruent Squares)
 Let the straight lines $AB$ and $LO$ be set out, and let $AB$ be greater, and let the semicircle $ABC$ have been drawn around it.
 And let $AC$, equal to the straight line $LO$, which is not greater than the diameter $AB$, have been inserted into the semicircle $ABC$ [Prop. 4.1].
 And let $CB$ have been joined.
 Therefore, since the angle $ACB$ is in the semicircle $ACB$, $ACB$ is thus a right angle [Prop. 3.31].
 Thus, the (square) on $AB$ is equal to the (sum of the) squares on $AC$ and $CB$ [Prop. 1.47].
 Hence, the (square) on $AB$ is greater than the (square) on $AC$ by the (square) on $CB$.
 And $AC$ (is) equal to $LO$.
 Thus, the (square) on $AB$ is greater than the (square) on $LO$ by the (square) on $CB$.
 Therefore, if we take $OR$ equal to $BC$ then the (square) on $AB$ will be greater than the (square) on $LO$ by the (square) on $OR$.
 (Which is) the very thing it was prescribed to do.
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"