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I added to my online research book the following theorem:

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

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

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