elem.5.19
若整体AB比整体CD等于减去的部分AE比减去的部分CF，则剩余部分EB比剩余部分FD也等于整体AB比整体CD。
若 AB:CD = AE:CF，则其差 EB:FD = AB:CD。下排 I、M、O、P、R、S 是引理里的辅助等倍量标号。
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For since, as AB is to CD, so is AE to CF, alternately also, as BA is to AE, so is DC to CF. [V. 16] And, since the magnitudes are proportional componendo, they will also be proportional separando, [V. 17] that is, as BE is to EA, so is DF to CF, and, alternately, as BE is to DF, so is EA to FC.
设AB:CD = AE:CF，由V.16交替得BA:AE = DC:CF。
[V. 16] But, as AE is to CF, so by hypothesis is the whole AB to the whole CD. Therefore also the remainder EB will be to the remainder FD as the whole AB is to the whole CD.
因合比成立，由V.17分比得BE:EA = DF:CF。
[V. 11] Therefore etc. [ PORISM.
再由V.16交替得BE:DF = EA:FC。
From this it is manifest that, if magnitudes be proportional componendo, they will also be proportional convertendo.
由假设AE:CF = AB:CD，故由V.11得EB:FD = AB:CD。