Question about chain complex in algebraic topology

[seriously disabled]

In algebraic topology there is something called a chain complex.

 

http://en.wikipedia.org/wiki/Chain_complex

 

My question is: Why is the composition of any two consecutive maps [math]d_n \cdot d_{n+1}[/math] equal zero?

[ajb]

You want [math]Im(d_{n+1}) \subseteq Ker(d_{n})[/math] so that the homology groups are well defined. This follows from the nilpotent property of the operator.