If $\baseof{L} = M$, trivially $D \isin M \equiv D \isin \baseof{L}.$
Whereas if $\baseof{L} = \baseof{M}$, by definition of $\base$,
$\patchof{M} = \patchof{L} = \py$, so by Tip Contents of $M$,
$D \isin M \equiv D \isin \baseof{M} \equiv D \isin \baseof{L}$.
If $\baseof{L} = M$, trivially $D \isin M \equiv D \isin \baseof{L}.$
Whereas if $\baseof{L} = \baseof{M}$, by definition of $\base$,
$\patchof{M} = \patchof{L} = \py$, so by Tip Contents of $M$,
$D \isin M \equiv D \isin \baseof{M} \equiv D \isin \baseof{L}$.