*Proof.* In the case when $k>M*$, for a single isolated downtime event $e\u21c0=(i,Td,d)$ with $d\u2264WSi(Td)$, consider the following sets:

$A={\Delta N\u22650:s.t.,\u2211k=1Nsi(k)=\u2211k=1Ns\u0303i(k;e\u21c0),\u2200N\u2265\Delta N+Td+d}$

$B={\Delta N\u22650:s.t.,si(N)=s\u0303i(N;e\u21c0),\u2200N\u2265\Delta N+Td+d}$

$C={\Delta N\u22650:s.t.,si(N)=sM*(N),s\u0303M*(N;e\u21c0)=s\u0303i(N;e\u21c0),\u2200N\u2265\Delta N+Td+d}$

$D={\Delta N\u22650:s.t.,\u2211j=M*+1ibj(N)=0,\u2200N\u2265\Delta N+Td+d}$