Composite of Bijections is Bijection

Theorem
Every composite of bijections is also a bijection.

Proof

 * As every bijection is also by definition an injection, a composite of bijections is also a composite of injections.

Every composite of injections is also an injection by Composite of Injections is an Injection.


 * As every bijection is also by definition a surjection, a composite of bijections is also a composite of surjections.

Every composite of surjections is also a surjection by Composite of Surjections is a Surjection.


 * As a composite of bijections is therefore both an injection and a surjection, it is also a bijection.