Complete the cases for proving Lemma $1$ using proofs similar to the ones provided in the image.

Can use the following axioms of HK:

A$1$ A$($B$$A$)$

A$2($A$($B$$C$))(($A$$B$)($A$$C$))$

A$3(-$A$-$B$)($B$$A$)$

As well as $$properties such as:

Reflexivity

Monotonicity

Transitivity

Compactness

Metatheoretic MP $($MMP$)$

Deduction Theorem $($DT$)$

Inductive Hypothesis $($IH$)$