\item
The soundness of natural deduction for propositional logic can be proven via a \emph{mathematical course-of-values induction on the length of the Natural Deduction proof}. Let $M(k)$ be the following assertion:

$M(k)\coloneqq$ „For all sequents $\phi_1,\phi_2,\dots,\phi_n\vdash \psi$ which have a proof of length $k$,
it is the case that $\phi_1,\phi_2,\dots,\phi_n\models \psi$  holds.”

Discuss the \textbf{induction step}
$M(1)\wedge M(2) \wedge \dots \wedge M(k-1) \rightarrow M(k)$ under the assumption that the \textbf{$\andi$} was applied as a final rule to prove the conclusion.