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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}
$M(1)\land M(2) \land \dots \land M(k-1) \imp M(k)$ under the assumption that the \textbf{$\nege$} was applied as a final rule to prove the conclusion.