I have re-defined filter rebase. Now it is defined for arbitrary filter $\mathcal{A}$ on some set $\mathrm{Base}(\mathcal{A})$ and arbitrary set $A$.
The new definition is: $\mathcal{A}\div A = \{ X\in\mathscr{P}A \mid \exists Y\in\mathcal{A}: Y\cap A\subseteq X \}$.
It is shown that for the special case of $\forall X\in\mathcal{A}:X\subseteq A$ the new definition is equal to the old definition that is $\mathcal{A}\div A = \{ X\in\mathscr{P}A \mid \exists Y\in\mathcal{A}: Y\subseteq X \}$.