(Condinitoinal) Law of Total Expecation

Let L be a random variable. Let K be an integer-valued random variable with possible values 0,1,2,3,...

Then 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).

Why is this true? I think this is probably related to the law of total expectation. I know that if X is a random variable and A is any event, then E(X)=E(X|A)P(A)+E(X|A')P(A') where A' is the complement of A.

But I still don't get why 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) is true. How can this be derived from the law of total expectation? I'm confused in particular about how to deal with the condinitional |K≥h on the left hand side. Does the law of total expectation still apply if the LHS is a CONDITIONAL expectation?

Hopefully someone can explain this.

Thank you!