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.
若一数是另一数的一部分，如同减数是被减数的一部分，则余数也是余数的相同部分，如同整体是整体的相同部分。
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For let the number AB be that part of the number CD which AE subtracted is of CF subtracted; I say that the remainder EB is also the same part of the remainder FD that the whole AB is of the whole CD. For, whatever part AE is of CF, the same part also let EB be of CG.
设数AB是数CD的一部分，如同AE是CF的一部分。
Now since, whatever part AE is of CF, the same part also is EB of CG, therefore, whatever part AE is of CF, the same part also is AB of GF.
取CG，使得EB是CG的相同部分，如同AE是CF的相同部分。
[VII. 5] But, whatever part AE is of CF, the same part also, by hypothesis, is AB of CD; therefore, whatever part AB is of GF, the same part is it of CD also; therefore GF is equal to CD. Let CF be subtracted from each; therefore the remainder GC is equal to the remainder FD.
由第七卷命题5，AB是GF的相同部分，如同AE是CF的相同部分；但假设AB是CD的相同部分，故GF等于CD。
Now since, whatever part AE is of CF, the same part also is EB of GC, while GC is equal to FD, therefore, whatever part AE is of CF, the same part also is EB of FD.
两边减去CF，得GC等于FD；因EB是GC的相同部分，如同AE是CF的相同部分，故EB是FD的相同部分，如同AB是CD的相同部分。