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Rest can be achieved by a brief period lying.
=Rest can be achieved by lying (for 省略) a brief period。
=Rest can be achieved by lying (for省略）a brief period (of time省略)
备注(1):lying 为动名词作介词by宾语。
备注(2):a brief period为时间状语修饰动名词lying。
备注(3):介词for 表示动作持续时间长度。即lying for a brief period躺一短暂的时间
a period of +表示时间的名词
This crisis might last for a long period of time. 这次危机可能持续很长时间。
He stayed here for a period of a few months. 他在这里停留了几个月时间。
The attempts to make changes in such a brief period of time will not yield results.
在这么短暂的时期内做出改变的尝试不会产生结果。