If a magnitude be the same multiple of a magnitude that a part subtracted is of a part subtracted, the remainder will also be the same multiple of the remainder that the whole is of the whole.
For let the magnitude / A B / be the same multiple of the magnitude / C D / that the part / A E / subtracted is of the part / C F / subtracted;
I say that the remainder / E B / is also the same multiple of the remainder / F D / that the whole / A B / is of the whole / C D /.
For, whatever multiple / A E / is of / C F /, let / E B / be made that multiple of / C G /.
Then, since / A E / is the same multiple of / C F / that / E B / is of / C G /,
[G.171]therefore / A E / is the same multiple of / C F / that / A B / is of / G F /.
But, by the assumption, / A E / is the same multiple of / C F / that / A B / is of / C D /.
Therefore / A B / is the same multiple of each of the magnitudes / G F /, / C D /;
therefore / G F / is equal to / C D /.
Let / C F / be subtracted from each;
therefore the remainder / C G / is equal to the remainder / F D /.
And, since / A E / is the same multiple of / C F / that / E B / is of / C G /,
and / C G / is equal to / F D /,
therefore / A E / is the same multiple of / C F / that / E B / is of / F D /.
But, by hypothesis,
/ A E / is the same multiple of / C F / that / A B / is of / C D /;
therefore / E B / is the same multiple of / F D / that / A B / is of / C D /.
That is, the remainder / E B / will be the same multiple of the remainder / F D / that the whole / A B / is of the whole / C D /.
