linear algebra - link between difference and product of inverse matrices

Let $A, E \in \mathbb{R}^{n \times n}$ matrices with rank $n$ (so non singular). Let $\lambda, \mu \in \mathbb{R}$ two different values which are both not eigenvalues of the pencil $(A,E)$. I am currently reading about the Loewner framework and I need following proposition.

$\dfrac{(\lambda E - A) ^ {-1}-(\mu E - A) ^ {-1}}{\mu - \lambda} = (\mu E - A)^{-1} E (\lambda E - A)^{-1}$

I tested it in matlab and it is correct, but I have no idea why this holds. The problem is that I don't know a proposition between the product of matrices and the difference of two matrices. Can someone help me?

Answer

Multiply the equation

$\dfrac{(\lambda E - A) ^ {-1}-(\mu E - A) ^ {-1}}{\mu - \lambda} = (\mu E - A)^{-1} E (\lambda E - A)^{-1}$ from the left by $\mu E-A$ and then from the right by $\lambda E-A$ and look what happens .....

Its your turn to make a valid proof from the above observation ...