Proof: Multiply by One
Let's prove the following theorem:
if not (c = 0), then (a ⋅ (1 / c)) ⋅ c = a
Proof:
Given
| 1 | not (c = 0) |
|---|
| # | Claim | Reason |
|---|---|---|
| 1 | (1 / c) ⋅ c = 1 | if not (c = 0), then (1 / c) ⋅ c = 1 |
| 2 | (a ⋅ (1 / c)) ⋅ c = a ⋅ ((1 / c) ⋅ c) | (a ⋅ (1 / c)) ⋅ c = a ⋅ ((1 / c) ⋅ c) |
| 3 | a ⋅ ((1 / c) ⋅ c) = a ⋅ 1 | if (1 / c) ⋅ c = 1, then a ⋅ ((1 / c) ⋅ c) = a ⋅ 1 |
| 4 | a ⋅ 1 = a | a ⋅ 1 = a |
| 5 | a ⋅ ((1 / c) ⋅ c) = a | if a ⋅ ((1 / c) ⋅ c) = a ⋅ 1 and a ⋅ 1 = a, then a ⋅ ((1 / c) ⋅ c) = a |
| 6 | (a ⋅ (1 / c)) ⋅ c = a | if (a ⋅ (1 / c)) ⋅ c = a ⋅ ((1 / c) ⋅ c) and a ⋅ ((1 / c) ⋅ c) = a, then (a ⋅ (1 / c)) ⋅ c = a |
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