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110% Proof

Theorem: All natural numbers are equal.

Proof: It is sufficient to show that A=B for any two natural numbers.

Also, it is sufficient to show that for all N > 0, if A and B are two natural numbers that satisfy the rule [MAX(A, B) = N] then A = B.

Now, use mathematical induction.

First the anchor step: If N = 1, then A and B, being natural numbers, must both equal 1 by the rule. So A = B.

Now, assume that the claim is true for some value K, and we have to show it is true for K+1. Consider natural numbers A and B with MAX(A, B) = K+1. But then MAX((A-1), (B-1)) = K, implying that (A-1) = (B-1) or A = B.

QED!