If a number be that part of a number, which a number subtracted is of a number subtracted, the remainder will also be the same part of the remainder that the whole is of the whole.
For let the number / A B / be that part of the number / C D / which / A E / subtracted is of / C F / subtracted;
I say that the remainder / E B / is also the same part of the remainder / F D / that the whole / A B / is of the whole / C D /.
For, whatever part / A E / is of / C F /, the same part also let / E B / be of / G C /.
Now since, whatever part / A E / is of / C F /, the same part also is / E B / of / G C /,
therefore, whatever part / A E / is of / C F /, the same part also is / A B / of / G F /. [G.257]
But, whatever part / A E / is of / C F /, the same part also, by hypothesis, is / A B / of / C D /;
therefore, whatever part / A B / is of / G F /, the same part is it of / C D / also;
therefore / G F / is equal to / C D /.
Let / C F / be subtracted from each;
therefore the remainder / G C / is equal to the remainder / F D /.
Now since, whatever part / A E / is of / C F /, the same part also is / E B / of / G C /, while / G C / is equal to / F D /,
therefore, whatever part / A E / is of / C F /, the same part also is / E B / of / F D /.
But, whatever part / A E / is of / C F /, the same part also is / A B / of / C D /;
therefore also the remainder / E B / is the same part of the remainder / F D / that the whole / A B / is of the whole / C D /.
