Each component which meets $X$ lies entirely within $Y$.
Then $C$ lies on no segment both of whose endpoints lie in $K$.
The advantage of using ...... lies in the fact that ......
This simple observation lies at the heart of the proof of our first theorem.
The importance of these examples lay not only in lowering the dimension of known counterexamples, but also in ...... [Note that the past tense of lie is lay, not “lied”.]
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