I added to my online research book the following theorem:

**Theorem** Let $latex \mathfrak{A}$ be a distributive lattice with least element. Let $latex a,b\in\mathfrak{A}$. If $latex a\setminus b$ exists, then $latex a\setminus^* b$ also exists and $latex a\setminus^* b=a\setminus b$.

The user quasi of Math.SE has helped me with the proof.