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