Let and be sub-manifolds of . Then and there are charts:

- for about adapted to
- for about adpapted to

such that

Now let be the injection map. Consider the commutative diagram below.

Then which is smooth. Hence is smooth.

Let be integral curves in a manifold then is closed.

**Proof:**

Let be have a limit in . Let be a chart then converges to by continuity. But also converges to . Hence and so . Thus is closed.