\[Definitions\] notation, $$1-\beta(\theta) 
= \sum_{k=1}^K \left(\prod_{j=1}^{k-1}\left[\beta_j(\theta)\right] \right) [1-\beta_k(\theta)]= 1 - \prod_{k=1}^{K}\beta_k(\theta).$$ At $K=1$, $\alpha_1$ and $n_1$ uniquely define $c_1$ and $\left\{T_{(1)} > c_{1}\right\}$ is a UPM by part 1 with a stage-specific power curve $1-\beta_1(\theta)$. At an arbitrary $K$, $\alpha_k$ and $n_k$ uniquely define $c_k$ and, by part 1 of the theorem, the stage-specific power $1-\beta_k(\theta)$, is the highest. Consequently, for any choice of $D$, the power of the test $\left\{T_{(D)} > c_{D}\right\}$ is $1-\beta(\theta) = 1-\prod_{k=1}^K\beta_k(\theta)$. This power is the highest, because the stage specific type 2 errors $\beta_k(\theta)$ are the lowest for each $k$ at any $\theta$. **Q.E.D.**\
Even if $T_k$ is normal, the distributions of $T_{(k)}$ are not normal, but Armitage’s recursive sub-density formula can be used to evaluate the distribution of $T_{(k)}$. For example, at $K=2$, the sub-density of $T_{(2)}$ is $f^{sub}_{T_{(2)}}(t|\theta) = \int_{-\infty}^{c_1} f_{T_{(2)}|T_{(1)}} (t|t_1,\theta) f_{T_{(1)}}(t_1|\theta)  dt_1$ and its density is $f_{T_{(2)}}(t|\theta) = f^{sub}_{T_{(2)}}(t|\theta)\left(\int_{-\infty}^{c_1} f_{T_{(1)}}(t_1|\theta)dt_1\right)^{-1}$. Recursively, the density conditional on reaching the $k^{th}$ interim analysis is $$f_{T_{(k)}}(t|\theta) = \frac{\int_{-\infty}^{c_{k-1}} f_{T_{(k)}|T_{(k-1)}}(t|t_{k-1},\theta) f_{T_{(k-1)}}(t_{k-1}|\theta)  dt_{k-1}}{\int_{-\infty}^{c_{k-1}} f_{T_{(k-1)}}(t_{k-1}|\theta)dt_{k-1}}.$$

This section has shown the impact of the possibility of early stopping on the finite sample distribution of test statistics. Section \[LSP\] shows how interim decisions to stop or continue affect asymptotic 