Generated Finite Partition of Generated Finite Sub-Sigma-Algebra is Itself

Theorem
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\eta$ be a finite partition of $\Omega$.

Then:
 * $\map \xi {\map \sigma \eta} = \eta$

where:
 * $\map \sigma \cdot$ denotes the generated $\sigma$-algebra
 * $\map \xi \cdot$ denotes the generated finite partition.