By definition, the lower left cell is green. Consistent coloring implies that any contour must be green if it lies below and to the left of the one passing through the upper right corner of this lower left cell [i.e., the contour through the points (0.04, 1), (0.2, 0.2), (1 0.04) in the numerical example in Table 4.5], since (a) it passes through the lower left cell (which is green by definition) and (b) none of the cells it passes through is red (by Lemma 2). By construction, such a green contour passes through all cells in the leftmost column and in the bottom row.

Now, consider the cell directly above the lower left cell [i.e., the cell containing the point (0.1, 0.3) in Table 4.5]. Suppose that, contrary to the claimed result, this cell is not green. It cannot be red, by Lemma 2. For it to be an intermediate color (not green), it must contain at least one red contour (by color consistency and the fact that a green contour passes through it. This cell cannot be “between” red and green cells, since it is on an edge of the matrix, so it cannot acquire an intermediate color that way.). This red color cannot come from the cell above it in the leftmost column (which is non-red, by Lemma 2), nor from any cell in the bottom row (again by Lemma 2). Since contours are downward-sloping, the only remaining possibility is that the cell to its right [the cell containing (0.3, 0.3) in Table 4.5] must be red. But this would violate betweenness [at the point (0.2, 0.2) in Table 4.5]. Therefore, the assumption that the cell directly above the lower left cell is not green leads to a contradiction. Hence, it must be green. By a symmetrical argument, the cell directly to the right of the lower left cell [the cell containing (0.3, 0.1) in Table 4.5] must also be green.

Next, suppose that the third cell in the leftmost column [the one containing (0.1, 0.5) in Table 4.5] is not green. Since green contours pass through it (as it is in the leftmost column), it can only be non-green if some red contour also passes through it (by color consistency and the fact that it is an edge cell). This red contour could not come from a red cell below it in the leftmost column, or in the bottom row (by Lemma 2), nor from the cell directly to its southeast [containing (0.3, 0.3) in Table 4.5] (since if that were red, it would violate Lemma 1 and betweenness for the cells so far proved to be green). The only remaining possibility is that the cell to its right [the one containing (0.3, 0.5) in Table 4.5] is red. But this would violate betweenness [with the second cell in the leftmost column, the cell containing (0.1, 0.3) in Table 4.5, which we have proved above must be green]. Hence, the assumption that the third cell in the leftmost column is not green implies a contradiction. So, it must be green. Symmetrically, the third cell in the bottom row must be green. This construction (showing that a cell directly above a green cell in the first column, with only non-red cells to its southeast, must itself be green) can be iterated for all remaining cells in the leftmost column, thus establishing that they all must be green; symmetrically, all remaining cells in the bottom row must be green. This proves part (a). Part (b) is then an immediate consequence of part (a) and Lemma 2. QED.

*Comment:* This proof does not depend on the number of rows or columns in the table. Therefore, its conclusion (that the leftmost column and bottom row consist entirely of green cells) holds for risk matrices of any size, under the stated conditions of weak consistency, betweenness, and consistent coloring.