If a number be the same parts of a number that a number subtracted is of a number subtracted, the remainder will also be the same parts of the remainder that the whole is of the whole.
For let the number / A B / be the same parts of the number / C D / that / A E / subtracted is of / C F / subtracted;
I say that the remainder / E B / is also the same parts of the remainder / F D / that the whole / A B / is of the whole / C D /.
For let / G H / be made equal to / A B /.
Therefore, whatever parts / G H / is of / C D /, the same parts also is / A E / of / C F /.
Let / G H / be divided into the parts of / C D /, namely / G K /, / K H /, and / A E / into the parts of / C F /, namely / A L /, / L E /;
thus the multitude of / G K /, / K H / will be equal to the multitude of / A L /, / L E /.
Now since, whatever part / G K / is of / C D /, the same part also is / A L / of / C F /, while / C D / is greater than / C F /,
therefore / G K / is also greater than / A L /.
Let / G M / be made equal to / A L /.
Therefore, whatever part / G K / is of / C D /, the same part also is / G M / of / C F /;
therefore also the remainder / M K / is the same part of the remainder / F D / that the whole / G K / is of the whole / C D /. [G.259]
Again, since, whatever part / K H / is of / C D /, the same part also is / L E / of / C F /, while / C D / is greater than / C F /,
therefore / K H / is also greater than / L E /.
Let / K N / be made equal to / L E /.
Therefore, whatever part / K H / is of / C D /, the same part also is / K N / of / C F /;
therefore also the remainder / N H / is the same part of the remainder / F D / that the whole / K H / is of the whole / C D /. [G.259]
But the remainder / M K / was also proved to be the same part of the remainder / F D / that the whole / G K / is of the whole / C D /;
therefore also the sum of / M K /, / N H / is the same parts of / F D / that the whole / G H / is of the whole / C D /.
But the sum of / M K /, / N H / is equal to / E B /, and / G H / is equal to / A B /;
therefore the remainder / E B / is the same parts of the remainder / F D / that the whole / A B / is of the whole / C D /.
