Region/Row 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 region as a possible value, then, that value may not appear anywhere else in the row (outside of its intersection with the region).

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 bottom row with the righthand region. The possible solution 3 appears twice in these three cells. The possible solution 3 does not appear in any other unsolved cell in this region. Therefore, one of these 3's must be the solution for this region. Therefore, in cells 1 & 2 of the bottom row, the possible solution of 3 may be eliminated.

To look at it another way, suppose that the solution to either cell 1 or cell 2 in the bottom row is 3. This eliminates 3 as a possible solution to any other cell in the row. This leaves the righthand region without a 3, which is clearly a nonsense.