Hilbert Proof System Instance 2 Independence Results

Theorem
Let $\mathscr H_2$ be Instance 2 of the Hilbert proof systems.

Then the following independence results hold:

Independence of $(A1)$
Axiom $(A1)$ is independent from $(A2)$, $(A3)$, $(A4)$.

Independence of $(A2)$
Axiom $(A2)$ is independent from $(A1)$, $(A3)$, $(A4)$.

Independence of $(A3)$
Axiom $(A3)$ is independent from $(A1)$, $(A2)$, $(A4)$.

Independence of $(A4)$
Axiom $(A4)$ is independent from $(A1)$, $(A2)$, $(A3)$.