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How to write output to an external text file in addition to the standard output stream?

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How to write output to an external text file in addition to the standard output stream, i.e. a notebook for a session with FE and stdout for command-line session?

Answers

I expected this to work but it apparently only writes Print output:

AppendTo[
  $Output,
  OpenWrite["out.txt", FormatType -> OutputForm]
]

A workaround using $PrePrint:

output = OpenWrite["out.txt", FormatType -> OutputForm];

$PrePrint = (Write[output, #]; #) &;

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