  We extend the combinatorial Morse complex construction to the arbitrary free
chain complexes, and give a short, self-contained, and elementary proof of the
quasi-isomorphism between the original chain complex and its Morse complex.
  Even stronger, the main result states that, if $C_*$ is a free chain complex,
and $\cm$ an acyclic matching, then $C_*=C_*^\cm\oplus T_*$, where $C_*^\cm$ is
the Morse complex generated by the critical elements, and $T_*$ is an acyclic
complex.
