Proportion of Numbers is Transitive

Theorem

 * If there be as many (natural) numbers as we please, and others equal to them in multitude, which taken two and two are in the same ratio, they will also be in the same ratio ex aequali.

Proof
Let there be as many (natural) numbers as we please, $A, B, C$, and others equal to them in multitude, $D, E, F$, which taken two and two are in the same ratio, so that:
 * $A : B = D : E$
 * $B : C = E : F$

We need to show that $A : C = D : F$.


 * Euclid-VII-14.png

We have that $A : B = D : E$.

So from, it follows that $A : D = B : E$.

Similarly, we have $B : C = E : F$.

So again from, it follows that $B : E = C : F$.

Putting them together, we get $A : D = C : F$.

Finally, again from, it follows that $A : C = D : F$.