  We give a new proof of the Morse Homology Theorem by constructing a chain
complex associated to a Morse-Bott-Smale function that reduces to the
Morse-Smale-Witten chain complex when the function is Morse-Smale and to the
chain complex of smooth singular $N$-cube chains when the function is constant.
We show that the homology of the chain complex is independent of the
Morse-Bott-Smale function by using compactified moduli spaces of time dependent
gradient flow lines to prove a Floer-type continuation theorem.
