Systematic component variation

#1
The appendix of the paper of [McPherson et al (1982) contains a derivation of the systematic component variation SCV. I understand the derivation with exception of the first step. Here are the premisses:

\(O_i\): observed cases in region i
\(E_i\): expected cases in region i
\(\lambda_i\): multiplicative factor associated with region i (I suppose it means \(O_i=\lambda_i*E_i\))

Now the following assumptions have been done:

\(O_i\) is approximately Poisson distributed with mean \(\lambda_iE_i\)
\(\lambda_i\) is considered as a random variable with expected value \(1\) and variance \(\sigma^2\).

From these the following formula is concluded:

var(\(O_i\)) = \(E_i^2\sigma^2\) + \(E_i\)


It tried to find out how to get the formula by the given premisses and assumptions and didn't succeed. Any idea? Thanks for help.