For a given event F, let PF (E) = P(E|F).

(a) Show that PF (E) satisfies the three axioms for being a probability function on events E.

(b) The identity PF (E) = PF (E|G)PF (G) + PF (E|Gc)PF (Gc) is seemingly equivalent to P(E|F) = P(E|FG)P(G|F) + P(E|FGc)P(Gc|F)
Give a direct proof of the preceding identity.