Theorem: there are 500,000 odd integers between zero and a million.
Proof #1: there ar no odd integers between zero and 1 (inclusive), 1 is odd so there is 1 odd integer between zero and 2, 2 is even so there is 1 odd integer between zero and 3, 3 is odd so there are 2 odd integers between zero and 4, …, 999,998 is even so there are 499,999 odd integers between zero and 999,999, 999,999 is odd so there are 500,000 odd integers between zero and 1,000,000. QED.
Proof #2: this is a specific case of “there are n odd integers between zero and 2n (exclusive)”. (proof of the more general theorem). Picking n to be 500,000, the theorem follows.
I think most people would call those two proofs different.
Proof #1: there ar no odd integers between zero and 1 (inclusive), 1 is odd so there is 1 odd integer between zero and 2, 2 is even so there is 1 odd integer between zero and 3, 3 is odd so there are 2 odd integers between zero and 4, …, 999,998 is even so there are 499,999 odd integers between zero and 999,999, 999,999 is odd so there are 500,000 odd integers between zero and 1,000,000. QED.
Proof #2: this is a specific case of “there are n odd integers between zero and 2n (exclusive)”. (proof of the more general theorem). Picking n to be 500,000, the theorem follows.
I think most people would call those two proofs different.