Jordan-Hölder Theorem

Theorem
Let $G$ be a finite group.

Let $\mathcal H_1$ and $\mathcal H_2$ be two composition series for $G$.

Then:
 * $\mathcal H_1$ and $\mathcal H_2$ have the same length
 * Corresponding factors of $\mathcal H_1$ and $\mathcal H_2$ are isomorphic.

Proof
By the Schreier-Zassenhaus Theorem, two composition series have refinements of equal length whose factors are isomorphic.

But from the definition of composition series, $\mathcal H_1$ and $\mathcal H_2$ have no proper refinements.

Hence any such refinements must be identical with $\mathcal H_1$ and $\mathcal H_2$ themselves.