Properties of resolvent operators

By definition,

$$R_{\lambda} (\lambda-A)= (\lambda-A) R_{\lambda} = \text{id}.$$

Consequently,

$$\begin{align*} R_{\lambda}-R_{\mu} &= R_{\lambda} (\mu-A) R_{\mu} - R_{\lambda}(\lambda-A) R_{\mu} \\ &= R_{\lambda} ((\mu-A)-(\lambda-A)) R_{\mu}. \end{align*}$$

This proves (12). In order to show (13) use (12). (Hint: What happens if we switch $\lambda \leftrightarrow \mu$ in (12)?)