Theorem: All positive integers are equal.
Proof: Sufficient to show that for any two positive integers, A and B, A = B.
Further, it is sufficient to show that for all N > 0, if A and B (positive integers) satisfy (MAX(A, B) = N) then A = B.
Proceed by induction.
If N = 1, then A and B, being positive integers, must both be 1. So A = B.
Assume that the theorem is true for some value k. Take A and B with MAX(A, B) = k+1. Then MAX((A1), (B1)) = k. And hence (A1) = (B1). Consequently, A = B.
