We prove this by strong induction relative to the expression \(2n - d\). Suppose that for all integers \(m\) and \(c\) with \(|2m - c|<|2n-d|\), it is true that -\(\operatorname{mod}(n, d) + d \cdot \operatorname{div}(n, d) = n\).
+\(\operatorname{mod}(m, c) + c \cdot \operatorname{div}(m, c) = m\).Case 1 (\(nd<0\)): Then by the inductive hypothesis