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\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.”
Your tasks: \begin{enumerate} \setlength{\itemsep}{-0.1em} \item Proof the induction base-case, i.e., $M(1)$ holds. \item Explain the proof idea of the induction step: $M(1)\wedge M(2) \land \dots \land M(k-1) \rightarrow M(k)$. \end{enumerate}
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