Row/Region elimination solve

First let's restate the basic premise. With this algorithm we are looking at the intersection of rows and regions. If, within that intersection, there are two or more unsolved cells and within those unsolved cells there is a possible value that doesn't appear in any other unsolved cell in the row as a possible value, then, that value may not appear anywhere else in the region (outside of its intersection with the row).

Why should this be so? The simplest way to show this is with an example. Suppose we have this situation for a set of three regions/rows.

Consider the intersection of the top row with the middle region. The possible solution 1 appears in all these three cells. The possible solution 1 does not appear in any other unsolved cell in this row. Therefore, one of these 1's must be the solution for this row. Therefore, in cells 4, 5 & 6 of the middle row, the possible solution of 1 may be eliminated.

To look at it another way, suppose that the solution to one of cell 4, cell 5 or cell 6 in the middle row is 1. This eliminates 1 as a possible solution to any other cell in the region. This leaves the top row without a 1, which is clearly a nonsense.