Theorem 44.4 and Theorem 44.7 are closely connected with Exercise 39 of Section 14. Show the connection. 

Data from Theorem 44.7

If A' is a subcomplex of the chain complex A, then the collection A/A' of factor groups A_k/A'k, together with the collection ∂̅ of homomorphisms ∂̅_k defined by ∂̅k(c + A'_k = ∂̅_k(c) + A'_k-1 for c ∈ A_k, is a chain complex. Since A/A' is a chain complex, we can then form the homology groups H_k(A/A').