How can I save the output of a pipe?

The shell lets us redirect the output of a sequence of piped commands:

cut -d , -f 2 seasonal/*.csv | grep -v Tooth > teeth-only.txt

However, > must appear at the end of the pipeline: if we try to use it in the middle, like this:

cut -d , -f 2 seasonal/*.csv > teeth-only.txt | grep -v Tooth

then all of the output from cut is written to teeth-only.txt, so there is nothing left for grep and it waits forever for some input.

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What happens if we put redirection at the front of a pipeline as in:

> result.txt head -n 3 seasonal/winter.csv