and thus the ordering of the columns of the X$_{s}$$_{ri}$ is now We proceed by induction over the number of stages completed l , for general D . Now, we assume that at the l th interim analysis, the statement of the Theorem is true. Now, the covariance at this l th analysis is It is this specifically we assume follows the required condition. Additionally, we assume that this covariance matrix can be computed, i.e. that A is invertible. We show that if we conduct another stage of the trial with t treatments remaining, 2 ≤ t ≤ q , then the new covariance matrix has the required property for ˆ π$_{2}$$_{l}$, . . . , ˆ π$_{tl}$, ˆ τ$_{1}$$_{l}$, . . . , ˆ τ$_{t}$$_{−}$$_{1}$$_{l}$ as well as for ˆ π$_{2}$$_{l}$, . . . , ˆ π$_{tl}$, ˆ τ$_{1}$$_{l}$, . . . , ˆ τ$_{t}$$_{−}$$_{1}$$_{l}$ and ˆ π$_{t}$$_{+1}$$_{l}$, . . . , ˆ π$_{Dl}$ , ˆ τ$_{tl}$, . . . , ˆ τ$_{D}$$_{−}$$_{1}$$_{l}$ . Let Denote dim( T$_{l}$ ) = | T$_{l}$ | , and similarly for T ′ l and W$_{l}$ . By our assumptions, we can write where for example A − 1 T$_{l}$T$_{l}$ = cov( T$_{l}$ , T$_{l}$ | L$_{l}$$_{1}$, . . . , L$_{lD}$ ), and with A − 1 T$_{l}$T$_{l}$ and A − 1 T ′ $_{l}$T$_{l}$ = ( A − 1 T$_{l}$T ′ l ) T holding the required conditions for the fixed effects. Finally, as part of our inductive hypothesis we also assume 27 2. For the next part of the theorem, we re-order our vector of fixed effects such that X$_{s}$$_{ri}$ = ( 1$_{r}$ Π$_{2}$$_{r}$ T$_{1}$$_{s}$$_{ri}$ Π$_{3}$$_{r}$ T$_{2}$$_{s}$$_{ri}$ . . . Π$_{Dr}$ T$_{D}$$_{−}$$_{1}$$_{s}$$_{ri}$$^{)}$ . cov( ˆ β$_{l}$ , ˆ β$_{l}$ | L$_{l}$$_{1}$ = · · · = L$_{lq}$$_{−}$$_{1}$ = 0 , L$_{lq}$, . . . , L$_{lD}$ ) =   D ∑ r = q L$_{r}$ n | S$_{r}$ | | S$_{r}$ | ∑ i =1 X T $_{s$_{ri}$}$Σ − 1 r X$_{s}$$_{ri}$$^{}$  − 1 = A − $^{1}$. cov( ˆ β$_{l}$$_{+1}$ , ˆ β$_{l}$$_{+1}$ | L$_{l}$$_{+11}$, . . . , L$_{l}$$_{+1}$$_{D}$ ) =  $_{}$A + n | S$_{t}$ | | S$_{t}$ | ∑ i =1 X T $_{s$_{ti}$}$Σ − 1 t X$_{s}$$_{ti}$$^{}$  − 1 = ( A + B ) − 1 , T$_{l}$ = (ˆ µ$_{0}$$_{l}$, ˆ π$_{2}$$_{l}$, ˆ τ$_{1}$$_{l}$, . . . , ˆ π$_{tl}$, ˆ τ$_{t}$$_{−}$$_{1}$$_{l}$ ) $^{T}$, T ′ l = (ˆ π$_{t}$$_{+1}$$_{l}$, ˆ τ$_{tl}$, . . . , ˆ π$_{Dl}$, ˆ τ$_{D}$$_{−}$$_{1}$$_{l}$ ) $^{T}$, W$_{l}$ = (ˆ π$_{2}$$_{l}$, ˆ τ$_{1}$$_{l}$, . . . , ˆ π$_{tl}$, ˆ τ$_{t}$$_{−}$$_{1}$$_{l}$ ) $^{T}$. A − 1 =    A − 1 T$_{l}$T$_{l}$ A − 1 T$_{l}$T ′ l A − 1 T ′ $_{l}$T$_{l}$ A − 1 T ′ $_{l}$T ′ l   $_{}$, β = ( µ$_{0}$, π$_{2}$, τ$_{1}$, π$_{3}$, τ$_{2}$, . . . , π$_{D}$, τ$_{D}$$_{−}$$_{1}$ ) $^{T}$,