Corollary 2

If two simple A-modules of finite dimension over K have a common nonzero coefficient, then they are isomorphic.

Let $S_1$ and $S_2$ be simple A-modules of finite dimension over K with a common nonzero coefficient. We then have $\Theta_{S_1}(A)\cap \Theta_{S_2}(A)\not= 0$. Now, $\Theta_{S_1}(A)$ is isotypical of type $S_1$, and $\Theta_{S_2}(A)$ is isotypical of type $S_2$. So $S_1$ is isomorphic to $S_2$.

By Theorem 1, $\Theta^{ss}(A)$ is an $(A$, A)-sub-bimodule of $A^*$, the direct sum of the $(A$, A)-bimodules $\Theta_S(A)$ for S running through $\mathscr{S}_K$. These bimodules are pairwise nonisomorphic because they are already nonisomorphic as left A-modules.

Fix an S in $\mathscr{S}_K$. Denote by D the opposite field of End$_A$(S), and view S as a right D-module and $S^*$ as a left D-module. With these conventions, S is an $(A$, D)-bimodule, $S^*$ is a $(D$, A)-bimodule, and $S\otimes_DS^*$ is an $(A$, A)-bimodule.