Suppose, I have conjectured some general law ("The longer the tail is, the cleverer is the owner", say). How to express the idea that I am going to confirm this law by considering a simple specific case (cats, say)?

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    "I'll show it with a simple example," or (more formally) "I'll show it by means of a simple example." – Mick Dec 20 '16 at 12:20
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    A minor correction: it sounds more fluent to say "The longer the tail is, the cleverer the owner is" so the structure is parallel in both parts of the sentence. – stangdon Dec 20 '16 at 18:15

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