# Difference cochain properties

I'm reading about obstruction theory. It's said that difference cochain $\delta (f_n,g_n)$ has properties:

$\delta(f_n,g_n)=0$ iff $f_n\simeq g_n (rel X_{n-1})$.

$\delta(f,g)-\delta(g,h)=\delta(f,h).$

$\delta(f,g)=-\delta(g,f).$

$d\delta(f,g)=c(g)-c(f)$.

I'm looking for book or lectures where these properties are proved. Also, detailed answers would be really helpful.