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\subsection*{Answer}
The soundness of the rule follows by mathematical induction
from the following observations:
 \[ \begin{noj}
    \M{\sigma}{P => []P} \;\; \equiv \;\;
     (\M{\sigma_{1}}{P} => (\A i \in Nat : \M{\sigma_{1+i}}{P}) \V{.7}
   \M{\sigma^{+i}}{P => P'} \;\; \equiv \;\;
      \M{\sigma_{1+i}}{P} => \M{\sigma_{2+i}}{P}
    \end{noj}
 \]
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