Therefore if GK be greater than HO, LN is greater than MP; and if equal, equal; and if less, less. [V. A.
Again, suppose that HO and MP are equimultiples of EB and FD.
Then, because AE is to EB as CF is to FD; [Hypothesis.
and that GK and LN are equimultiples of AE and CF, and HO and MP are equimultiples of EB and FD;
therefore if GK be greater than HO, LN is greater than MP; and if equal, equal; and if less, less; [V. Definition 5.
which was likewise shewn on the preceding supposition.
But if GH be greater than KO, then by taking the common magnitude KH from both, GK is greater than HO;
therefore also LN is greater than MP;
and, by adding the common magnitude NM to both, LM is greater than NP.
Thus if GH be greater than KO, LM is greater than NP.
In like manner it may be shewn, that if GH be equal to KO, LM is equal to NP; and if less, less.
And in the case in which KO is not greater than KH, it has been shewn that GH is always greater than KO and also LM greater than NP.
But GH and LM are any equimultiples whatever of AB and CD, and KO and NP are any equimultiples whatever of BE and DF, [Construction.
therefore AB is to BE as CD is to DF. [V. Definition 5. Wherefore, if magnitudes &c. q.e.d.
