When p+ 瓦w-（4.五) V>2+E) nt2a干2a y(m-an-劲>Em+1n-3的+E(m-ajm+11. New Proof of Dimension Formula of Spline Spaces over T-meshes via Smoothing Cofactors 9 When p > 2q + 1, Ap,q(x, y) = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ xq+1 yq+1 . . . ... . . . ... xq+1 yq+1 . . . ... . . . ... . . . ... . . . ... 1 1 ... . . . ... ... ... . . . ... ... 1 ··· ··· xq+1 1 ··· ··· yq+1 ... . . . ... . . . ... . . . ... . . . 1 1 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ . Delete the first p − 2q − 1 columns of matrix Ap,q(y) and denote the result matrix by Ap,q(y), then matrix Ap,q =  Ap,q(x) Ap,q(y)  = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ xq+1 . . . ... xq+1 yq+1 . . . ... . . . ... . . . ... 1 ... . . . ... ... . . . ... ... 1 ··· ··· xq+1 1 ··· ··· yq+1 ... . . . ... . . . ... . . . ... . . . 1 1 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ is a (p + 1) × (p + 1) square matrix, it is easy to know the determinant det Ap,q = x(q+1)(p−2q−1)res(f, g, z) = x(q+1)(p−2q−1)(x − y) (q+1)2 = 0, hence, rank Ap,q(x, y) = p + 1. It is impossible for regular T-meshes to have only in-lines, since the number of adjacent edges of an interior vertex should be 3 or 4, corresponding to T-junctional and crossing vertices respectively. Furthermore, the endpoints of an in-line can’t be endpoints of other in-lines, but can be inner vertices of others. Then we have V > 2(Ei h + Ei v), where V is the number of interior vertices, Ei h and Ei v the number of horizontal in-lines and the number of vertical in-lines, respectively. So when m 2α + 1, n 2β + 1, it follows that V (m − α)(n − β) > Ei h(m + 1)(n − β) + Ei v(m − α)(n + 1). Hence, the coefficient matrix M in Equation(13) has more columns than rows. And when m 2α + 1, n 2β + 1, we have