[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Prove that for any three events [math]A[/math], [math]B[/math], [math]C[/math], each having positive probability, and with the property that [math]P(A \cap B) \gt 0[/math],

[[math]] P(A \cap B \cap C) = P(A)P(B|A)P(C|A \cap B)\ . [[/math]]