| ⊢ fin w ≡ w | BfFinStateEqvBox | 
Proof:
| 1 |   fin w ≡fin w          | |
| 2  |  fin w ≡fin w          | |
| 3  | finite  ⊃ (( fin w) ≡ w) | |
| 4  |   fin w ≡ w             | |
| 5  |   w ≡ w                | |
| 6  |  w ≡ w                  | |
| 7  |   w ≡ w                 | |
| 8  |  fin w ≡ w              | 
qed
An alternative proof of Theorem BfFinStateEqvBox can be given without PITLF by first deducing the dual equivalence ( (empty ∧ w)) ≡ w, for any state formula w.