Stefano Franchi wrote:

> that solves the problem. Can I ask you to explain what you did, so I can
> understand the solution and replicate it myself?

Sure.

> In the first solution, what did you insert after the beginning of the
> quote environment and before the beginning of the enumerate environment?
> All I see in LyX is an empty red rectangle, and the LaTex source window
> shows a completely empty line.

I have inserted an empty ERT into a quote environment (this is the red 
rectangle). Its purpose is only to keep the environment (if the quote 
paragraph was empty, LyX would just delete it). A note inset would probably 
do as well. This generates
\begin{quote}

In the next part, I inserted the enumeration, and I nested it into the quote 
environment (Edit>Increase List Depth). The purpose was to have the 
enumeration environment nested into the "empty" quote environment. I.e.,

\begin{quote}
\begin{enumeration}
\item foo
\item bar
\end{enumerate}
\end{quote}

> In the second solution, I understand the \setcounter part, but what is the
> meaning  of the open and close brackets as ERT?

This is the optional argument of the enumeration item, in LaTeX

\item[``1.] foo

In the optional argument, you can set an alternative enumeration label, so 
the label is ``1. instead of just 1.
The setcounter part is needed, because the counter needs to be increased 
(otherwise, the second item would start with 1.

Regards,
Jürgen


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