Kleene equality

In mathematics, Kleene equality, or strong equality, is an equality operator on partial functions, that states that on a given argument either both functions are undefined, or both are defined and their values on that arguments are equal.
For example, if we have partial functions and, means that for every :

and are both defined and
or and are both undefined.

Some authors are using "quasi-equality", which is defined like this:
where the down arrow means that the term on the left side of it is defined.
Then it becomes possible to define the strong equality in the following way: