It is shown that every holomorphic function on a nuclear Frechet space $E$ with values in a Frechet space $F$ is of uniform type if
$E$ has the linear topological invariant $(\tilde{\Omega})$ and $F$ has the linear topological invariant $(DN)$ respectively. Based on the obtained result the equivalence of the holomorphicity and the weak holomorphicity of Frechet-valued functions on $\tilde{L}$-regular compact subsets in a nuclear Frechet space is established.