# A question about the proof of Extension Lemma for Smooth functions

I am trying to understand the proof, but am stuck at the last line. Why does the first equality of the picture below holds? The codomain of $f$ is $R^k$, not some the set of nonnegative numbers. Is there something I am missing? I need hekp.

• It might be a closure of a set. – Atbey Jan 20 '18 at 9:16
• What do you mean by "it"? Could you explain in more detail? – Keith Jan 20 '18 at 9:18
• Why did you mention nonnegative numbers? – Atbey Jan 20 '18 at 9:27
• Because, nonzero elements in $R^k$ can add up to zero. So, there is no guarantee that the inclusion of the right side into left side on the first equality holds. – Keith Jan 20 '18 at 9:36
• You should mention the reference. – Sou Jan 20 '18 at 18:07