Logic-System Of Inference

I'll be delighted to get some help in the following:

Let D be the following system of inference: The axioms are all the tautologies except the ones that is their main symbol and the only rule of inference if the rule:

.

A. Prove that D is entire in the weak meaning, but not entire (I've proved it).

B. Let D' be the system of inference that derives from D by changing the rule by :

. Prove that D' is entire.

C. Sketch a proof tree for : from the set .

I've proved A... I realy need help in B and C.

Here is what I know:

Quote:

in part B: if we can prove somehow that D' is entire in the weak meaning, we will be able to deduce what we need (because of the special rule of inference we have) ... But how can we prove it?

In C-I know how to prove that

is true in D'... But can't figure out how to prove the full statement...

Thanks !