Proof: Add Three
Let's prove the following theorem:
(a + a) + a = a ⋅ 3
Proof:
| # | Claim | Reason |
|---|---|---|
| 1 | a + a = a ⋅ 2 | a + a = a ⋅ 2 |
| 2 | a = a ⋅ 1 | a = a ⋅ 1 |
| 3 | (a + a) + a = (a ⋅ 2) + a | if a + a = a ⋅ 2, then (a + a) + a = (a ⋅ 2) + a |
| 4 | (a ⋅ 2) + a = (a ⋅ 2) + (a ⋅ 1) | if a = a ⋅ 1, then (a ⋅ 2) + a = (a ⋅ 2) + (a ⋅ 1) |
| 5 | (a ⋅ 2) + (a ⋅ 1) = a ⋅ (2 + 1) | (a ⋅ 2) + (a ⋅ 1) = a ⋅ (2 + 1) |
| 6 | (a ⋅ 2) + a = a ⋅ (2 + 1) | if (a ⋅ 2) + a = (a ⋅ 2) + (a ⋅ 1) and (a ⋅ 2) + (a ⋅ 1) = a ⋅ (2 + 1), then (a ⋅ 2) + a = a ⋅ (2 + 1) |
| 7 | 2 + 1 = 3 | 2 + 1 = 3 |
| 8 | a ⋅ (2 + 1) = a ⋅ 3 | if 2 + 1 = 3, then a ⋅ (2 + 1) = a ⋅ 3 |
| 9 | (a ⋅ 2) + a = a ⋅ 3 | if (a ⋅ 2) + a = a ⋅ (2 + 1) and a ⋅ (2 + 1) = a ⋅ 3, then (a ⋅ 2) + a = a ⋅ 3 |
| 10 | (a + a) + a = a ⋅ 3 | if (a + a) + a = (a ⋅ 2) + a and (a ⋅ 2) + a = a ⋅ 3, then (a + a) + a = a ⋅ 3 |
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