This is a completely tedious proof. I will give you the outline.
But please no questions about it.
Suppose that .
That means that .
From which it follows that .
It follows from that .
Because both are transitive we get .
But that means that .
I have this problem that is really driving me crazy:
Suppose R and S are transitive relations on A. Prove that if SēR c RēS, then RēS is transitive.
I tried to use the theorem that says: R is transitive iff RēR c R, but I couldn't get to the end.
(c means subset)