If, on the other hand, the degree of the GCD is "i", then Bézout's identity again allows proving that the multiples of the GCD that have a degree lower than "m" + "n" − "i" are in the image of formula_42. The vector space of these multiples has the dimension "m" + "n" − 2"i" and has a base of polynomials of pairwise different degrees, not smaller than "i". This implies that the submatrix of the "m" + "n" − 2"i" first rows of the column echelon form of "Ti" is the identity matrix and thus that "si" is not 0. Thus "Si" is a polynomial in the image of formula_42, which is a multiple of the GCD and has the same degree. It is thus a greatest common divisor.