# Proof that for a strongly continuous contraction resolvent, there is exactly one linear operator that generates the resolvent.

I have questions about the proof of the following Proposition from the book Introduction to the Theory of Non-Symmetric Dirichlet forms.

First, how do we get the independence of $$G_\alpha (B)$$ of $$\alpha$$ by 1.4(iii) and that each $$G_\alpha$$ is one-to-one by 1.4(i) and 1.4(iii)?

Second, once we conclude that $$\alpha - G_\alpha^{-1} = \beta - G_\beta^{-1}$$, how do we get that $$D(L)=G_\alpha (B)$$ is dense in $$B$$ by 1.4 (i)?

I would greatly appreciate any help.