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來源: 於 2014-07-11 10:18:08 [博客] [舊帖] [給我悄悄話] 本文已被閱讀：次

Theorem 2: Commutativity of addition: a+b=b+a.

Proof: Base case: By lemma 1, a+1=1+a.

Inductive hypothesis: Suppose that, for k'>1, a+k=k+a.
By axiom 6, a+k'=a+(k+1).
By theorem 1, a+k'=(a+k)+1.
By the inductive hypothesis, a+k'=(k+a)+1.
By theorem 1, a+k'=k+(a+1).
By lemma 1, a+k'=k+(1+a).
By theorem 1, a+k'=(k+1)+a.
By axiom 6, a+k'=k'+a.
By induction, a+b=b+a. QED.