I base this upon an analogy between formal proof and causal consequences.

The realizer, however, usually gives more information about the formula than a formal proof would directly provide.

One example of such a formal proof is as follows.

Exploring a possible result to see if it is worth formal proof.

A formal proof of this is often an exercise in theoretical computer science courses.

The formal proof is somewhat technical and is beyond the current state of this article.

The field of proof theory studies formal proofs and their properties.

They are then believed to be hard to break, but no such formal proof is given.

A formal proof based on mathematical analysis would eliminate these potential ambiguities.

A formal proof of the powerset construction is given here.