I'm currently reading Loopholes in Bell Inequality Tests of Local Realism by Jan-Ake Larsson

<https://arxiv.org/abs/1407.0363>

On Page 6, Equation 7, he has a short proof, where I am having a hard time seeing through the math. I'll re-state here for convenience:

$ \lvert E(A\_{2}B\_{1}) - E(A\_{2}B\_{2}) \rvert = \lvert E(A\_{2}B\_{1} + A\_{2}B\_{1}A\_{1}B\_{2}) \rvert \leq E(\lvert A\_{2}B\_{1}(1 + A\_{1}B\_{2}) \rvert ) = 1 + E(A\_{1}B\_{2}) $

where 

$ A\_{1}B\_{1} = -1 $

and

$E(A) = \int A( \lambda ) \rho ( \lambda) d \lambda $

I'm not seeing how he goes from the 2nd expression on the LHS of the less-than-or-equal sign to the third expression on the RHS of the less-than-or-equal sign. I suspect I am missing something on the properties of expectation values. *Mainly, I think the absolute signs (|) moving from outside of the E()'s to the inside of the E()'s have me the most confused.* Further, I don't see how he goes from the 3rd expression to the 4th, either.

Can anyone offer any clarification here?