Proof of the Sobolev Extension Theorem in Evans' PDE

Question

From PDE Evans, pages 254-256(1st edition) or pages 268-270(2nd edition). In equation(9) of step 6 , Evans yields : $$ || \bar{u} ||_{W^{1,p \,}(R^n)} \leqslant C ||u||_{W^{1,p \,}(U)} $$ My question is : $$ \bar{u} := \sum_{i=0}^{N} \xi_i \bar{u}_i $$is defined in $ \cup_{i=0}^N W_i \,\ , $ and has no meaning outside this domain. i.e. This is not an extension to $R^n$. I can understand that $$ || \bar{u} ||_{W^{1,p \,}(\,\ \cup_{i=0}^N \,\ W_i)} \leqslant C ||u||_{W^{1,p \,}(U)} $$ hold true, but why the domain here can be replaced with $R^n$?