I am going through Milnor and Moore - On the structure of Hopf algebras (MSN) (I have already posted one question on that, another one is coming). My question is about Proposition 4.9, more specifically about the proof. It is stated that it is in parts already proved, but I am struggling with seeing what is going on in the proof that a) implies b), namely that:

Let $A$, $B$, $C$ be connected Hopf algebras over $K$ and $i:A\to B$, $\pi :B\to C$ be morphisms of Hopf algebras. Then a) implies b), where:

a) $i$ is a left normal monomorphism of algebras, $C=K\otimes_A B$ and the sequences $0\to A\to B$, $B\to C\to 0$ are split exact as sequences of graded $K$-modules,

b) $\pi$ is a right normal epimorphism of coalgebras, $A\Box_C K$, and the sequences as above are split exact as sequences of graded $K$-modules.

I will be very grateful for any help in seeing how the implication is proven.