From each of these take the common magnitude AE; then the remainder AG is equal to the remainder EB;
Then, because AE is the same multiple of CF that AG is of ED, [Construction. and that AG is equal to EB;
therefore AE is the same multiple of CF that EB is of ED.
But AE is the same multiple of CF that AB is of CD; ['Hypothesis.
therefore EB is the same multiple of FD that AB is of CD.
Wherefore, if one magnitude &c q.e.d.
PROPOSITION 6. THEOREM.
If two magnitudes he equimultiples of two others and if equimultiples of these he taken from, the first two, the remainders shall he either equal to these others, or equi-multiples of them.
Let the two magnitudes AB, CD be equimultiples of the two E, F; and let AG, CH, taken from the first two, be equimultiples of the same E, F: the remainders GB', HD shall be either equal to E, F, or equimultiples of them.
First, let GB be equal to E: HD shall be equal to F.
Make CK equal to F.
Then, because AG the same multiple of E that CH is of F, [Hyp.
and that GB is equal to E, and CK is equal to F;
therefore AB is the same multiple of E that is CH is of F.
But AB is the same multiple of E that CD is of F; [Hypothesis.
therefore KH is the same multiple of E that CD is of F;
therefore KH is equal to CD. [V. Axiom 1.
From each of these take the common magnitude CH; then the remainder CK is equal to the remainder HD. But CK is equal to F; [Construction.
therefore HD is equal to F.
