Show that the string $( [ ) ]$ is not in a Dyck language
Consider producing a leftmost derivation of ( [ ) ]. We can begin with 
$$
S\Rightarrow SS\stackrel{*}{\Rightarrow}SS\dotsc S
$$
and to get the ( on the left we must eventually use the production $S\rightarrow(S)$ on the leftmost $S$, giving 
$$
S\stackrel{*}{\Rightarrow}(S)S\dotsc S
$$
and then we'll have 
$$
S\stackrel{*}{\Rightarrow}(SS\dotsc S)S\dotsc S
$$
and to get the [ as the second terminal on the left we must eventually use the production $S\rightarrow[S]$, giving 
$$
S\stackrel{*}{\Rightarrow}([S]SS\dotsc S)S\dotsc S\stackrel{*}{\Rightarrow}([SS\dotsc S]SS\dotsc S)S\dotsc S
$$
and now we're stuck, since no production will give us the ) terminal we need on the left without introducing a ( first.

For a slightly different take, consider that $S$ can only derive strings in $D_2$, so to  derive ( ]  ) ] we must use the production $S\rightarrow (S)$, where the $S$ in the right term of the production must also derive a string in $D_2$, but that would imply $S$ derives ], which cannot be, since $]\notin D_2$.