> how do you verify the verification program?

The program used to check the validity of a proof is called a kernel. It just need to check one step at a time and the possible steps can be taken are just basic logic rules. People can gain more confidence on its validity by:

- Reading it very carefully (doable since it's very small)

- Having multiple independent implementations and compare the results

- Proving it in some meta-theory. Here the result is not correctness per se, but relative consistency. (Although it can be argued all other points are about relative consistency as well.)