**Proposition:** The product of functions is a function.

**Proof:**

- Let and both be functions.
- Let and .
- Then there exists such that and ;
- And there exists such that and .
- Since is a function, then .
- Since is a function, then .
- Therefore, is a function.

See the Definition of a Function in the category SET.

*Now go lift something heavy,*

Nick Horton

PS. Pic-credit Chris Huh.