It can be projectivized, as in the previous example, to obtain the projective Moulton plane. Desargues' theorem is not a valid theorem in either the Moulton plane or the projective Moulton plane. This example has just thirteen points and thirteen lines. We label the points P1...,P13 and the lines m1...,m13. The incidence relation (which points are on which lines) can be given by the following incidence matrix. The rows are labelled by the points and the columns are labelled by the lines. A 1 in row "i" and column "j" means that the point P"i" is on the line m"j", while a 0 (which we represent here by a blank cell for ease of reading) means that they are not incident. The matrix is in Paige-Wexler normal form. To verify the conditions that make this a projective plane, observe that every two rows have exactly one common column in which 1's appear (every pair of distinct points are on exactly one common line) and that every two columns have exactly one common row in which 1's appear (every pair of distinct lines meet at exactly one point). Among many possibilities, the points P1,P4,P5,and P8, for example, will satisfy the third condition. This example is known as the projective plane of order three. Though the line at infinity of the extended real plane may appear to have a different nature than the other lines of that projective plane, this is not the case. Another construction of the same projective plane shows that no line can be distinguished (on geometrical grounds) from any other. In this construction, each "point" of the real projective plane is the one-dimensional subspace (a "geometric" line) through the origin in a 3-dimensional vector space, and a "line" in the projective plane arises from a ("geometric") plane through the origin in the 3-space. This idea can be generalized and made more precise as follows. Let "K" be any division ring (skewfield). Let "K"3 denote the set of all triples "x" = ("x"0, "x"1, "x"2) of elements of "K" (a Cartesian product viewed as a vector space). For any nonzero "x" in "K"3, the minimal subspace of "K"3 containing "x" (which may be visualized as all the vectors in a line through the origin) is the subset of "K"3.