 Note now, that in determining the signs of the members of the sequence of polynomials formula_160 that at formula_161 the dominating power of formula_162 will be the first term of each of these polynomials, and thus only these coefficients corresponding to the highest powers of formula_162 in formula_164, and formula_138, which are formula_166, formula_167, formula_141, formula_149, ... determine the signs of formula_130, formula_131, ..., formula_138 at formula_173.So we get formula_174 that is, formula_175 is the number of changes of sign in the sequence formula_176, formula_177, formula_178, ... which is the number of sign changes in the sequence formula_166, formula_167, formula_141, formula_149, ... and formula_183; that is formula_184 is the number of changes of sign in the sequence formula_185, formula_186, formula_187, ... which is the number of sign changes in the sequence formula_166, formula_189, formula_141, formula_191, ...Since our chain formula_166, formula_167, formula_141, formula_149, ... will have formula_32 members it is clear that formula_197 since within formula_198 if going from formula_166 to formula_167 a sign change has not occurred, within 
formula_201 going from formula_166 to formula_189 one has, and likewise for all formula_32 transitions (there will be no terms equal to zero) giving us formula_32 total sign changes.