, Using the above characterization of a : b < < ? c : d = 1, we have different cases to consider

, Case a = b: in that case, a : b < < ? e : f = 1 obviously holds

Since (c : d < < ? e : f = 1), we necessarily have | d ? c | ? | f ? e | and e ? f . We deduce | b ? a | ? | f ? e | and a ? b and e ? f ,

, a similar reasoning applies to show that still a : b < < ? e : f = 1. ? Moreover a : b < < ? c : d = 0 if and only if

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