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