Is any anagram of the empty language the same as the empty language?

Question

Given a regular expression r, r~ contains all anagrams of r.

L(r) is the language accepting all words that can be constructed from r.

E.g. Given the language L(r) = {dog}, L(r~) = {dog,god,odg,ogd,dgo,gdo}.

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Given O which contains no words, L(O) is the empty language which contains nothing.

Question: Is the language L(O) = L(O~) ?

Answer 1

Yes? If $L(O) = \{\}$, then $L(O$~$) = \{$words that are an anagram of some word w in L(O)$\} = \{$nothing because there are no words w in $L(O)\}=\{\}$.