It is shown that $H(K, F)$ is regular for every reflexive Fréchet space $F$ with the property ($\mathrm{LB}_\infty)$ where $K$ is a compact set of uniqueness in a Fréchet-Schwartz space $E$ such that $E \in (\Omega)$. Using this result we give necessary and sufficient conditions for a Fréchet space $F$, under which every separately holomorphic function on $K \times F^*$ is holomorphic, where $K$ is as above.
