  Extension of two known facts concerning subordination is made. The first fact
is that, in subordination of 1-dimensional Brownian motion with drift,
selfdecomposability is inherited from subordinator to subordinated. This is
extended to subordination of cone-parameter convolution semigroups. The second
fact is that, in subordination of strictly stable cone-parameter convolution
semigroups on $\mathbb{R}^d$, selfdecomposability is inherited from
subordinator to subordinated. This is extended to semi-selfdecomposability.
