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