Questions about C1-diffeomorphisms

I proof the global version of the inverse function theorem using the following local version:

Inverse function theorem [local]. Let E⊂RnE⊂Rn open, and let f:E→Rnf:E→Rn be a C1C1 map. Let a∈Ea∈E be such that det(f′(a))≠0det(f′(a))≠0. Then there exists open balls U∋aU∋a and V∋f(a)V∋f(a) such that f:U→Vf:U→V is a C1C1 diffeomorphism.

I will proof:

Inverse function theorem [global] Let E⊂RnE⊂Rn open, and let f:E→Rnf:E→Rn be an injective 

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