That’s not a proof; it’s an (invalid) inference — doubly invalid if you have no form for imperfect induction, but in any case invalid by the introduction of ‘we know that’ as a modal operator (are we doing epistemic logic? p->q |= p->Kq isn’t a valid schema).
It’s not in FOL, because of the modal operator, and it’s an instance of a schema, not a proof. Not what I asked for.
Anyway, I wrote something on Maps of Meaning in case you’re interested.