For natural numbers with zero m ≤ n, the difference n − m is the unique natural number with zero that added to m gives n.
In the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, let with .
The difference is the unique with ; such a exists by Arithmetic of Addition on Omega: Recursion Rules, Associativity, Commutativity, Cancellation and Compatibility with the Order §difference and is unique by Arithmetic of Addition on Omega: Recursion Rules, Associativity, Commutativity, Cancellation and Compatibility with the Order §cancellation.
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