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