I think the multiplication law in probability is the easiest proposition to prove in the world.
Why is the multiplication law useful?
Multiplication Law:
Let $A$ and $B$ be events and assumu $P(B)\ne 0$. Then $$P(A\cap B) = P(A|B) P(B).$$
My proof of this proposition is here:
By definion, $P(A|B) := \frac{P(A\cap B)}{P(B)}$, so $P(A|B) P(B) = \frac{P(A\cap B)}{P(B)} P(B) = P(A\cap B)$.
Q.E.D.
I think this proposition is too easy to be useful.
Why is the multiplication law useful?
By the way, I asked a similar question before, but I was not able to understand the answer. (About Conditional Probability, Circular Argument?)