E(L|K≥h) = E(L|K≥h+1)P(K≥h+1|K≥h) + E(L|K<h+1)P(K<h+1|K≥h).

Now I've only seen the theorem E(X)=E(E(X|Y), bascially I'm wondering if this would work if the left hand side is a CONDITIONAL expectation like the above.

Since there is a |K>=h on the left hand side, can we just simply put |K>=h EVERYWHERE on the four quantities on the right hand side as well?

Thanks.