Thomas decomposition and its applications

All zeroes of inequations have to be removed from the solution set. This requires them to be treated similarly to the equations. For example, consider the system S := {p = 0, q 6= 0} = {x − x + 1 = 0, x + a 6= 0}. For almost all values a, S|a=a has two distinct zeroes, exceptions determined by r(a) = 0, where r = res(p, q, x) = a + a + 1. Use r to decompose S into the simple systems {p = 0, r 6= 0} and {x − a − 1 = 0, r = 0}.