For the mathematically inclined, the relation that defines the natural ordering on a given class C is:
{(x, y) such that x.compareTo(y) <= 0}.
The quotient for this total order is:
{(x, y) such that x.compareTo(y) == 0}.
It follows immediately from the contract for compareTo that the quotient is an equivalence relation on C, and that the natural ordering is a total order on C. When we say that a class's natural ordering is consistent with equals, we mean that the quotient for the natural ordering is the equivalence relation defined by the class's equals(Object) method:
{(x, y) such that x.equals(y)}.
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