pontrjagin ring of the homology of iterated loop suspension

Question

In The homology of C n+1 spaces, n>=0, F. Cohen, proof of Theorem 3.1 and proof of Theorem 3.2 (p. 228 - 243)

I totally do not understand the proofs of these two theorems from page 228 to page 243 in Section 3 and Section 4 of The homology of C n+1 spaces, n>=0. Could you explain the ideas for the proof, especially for the proof of Theorem 3.2?