Page "Nim" Paragraph 115

Proof: If there is no possible move, then the lemma is vacuously true ( and the first player loses the normal play game by definition ).

Otherwise, any move in heap k will produce t = x < sub > k </ sub > ⊕ y < sub > k </ sub > from (*).

This number is nonzero, since x < sub > k </ sub > ≠ y < sub > k </ sub >.