Proportion of Numbers is Transitive

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 $A : C = D : F$.