In this subsubsection, an example is given for a condition for a non-unital ring to be unital. This example is already mentioned in the comment for df-subrg: " The subset of (where multiplication is componentwise) contains the false identity which preserves every element of the subset and thus appears to be the identity of the subset, but is not the identity of the larger ring."
The theorems in this subsubsection do not assume that is a ring (which can be proven directly very easily, see pzriprng), but provide the prerequisites for ring2idlqusb to show that is a unital ring, and for ring2idlqus1 to show that is its ring unity.