Why is this theorem important in fulton's book?

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In Fulton algebraic curves book, we have the following proprieties which help us to find the intersection number of a pair of function at a given point.

afterwards he states this theorem:

What's the advantage of this characterization? I think it's simpler to use the proprieties of the intersection numbers to find them instead use this theorem.

Anyone have a simple example how this theorem can help me?

Thanks in advance