For any real number c, define c (ceiling) as an integer d such that z(0 z < 1) and d = c + z. Assume, as a premise, that c exists for every real c. For example for c = 2.34 we have d = 3 and z = 0.66 because 3 = 2.34 + 0.66. Prove that a, b : a + b is equal to a + b or a + b 1 where a = a + x for 0 x < 1 and b = b + y for 0 y < 1 by the definition of ceiling