Proof: Cross Multiply Theorem
Let's prove the following theorem:
if a / b = c / d, then a ⋅ d = b ⋅ c
Proof:
Given
| 1 | a / b = c / d |
|---|
| # | Claim | Reason |
|---|---|---|
| 1 | (a / b) ⋅ b = (c / d) ⋅ b | if a / b = c / d, then (a / b) ⋅ b = (c / d) ⋅ b |
| 2 | (a / b) ⋅ b = a | (a / b) ⋅ b = a |
| 3 | a = (c / d) ⋅ b | if (a / b) ⋅ b = a and (a / b) ⋅ b = (c / d) ⋅ b, then a = (c / d) ⋅ b |
| 4 | a ⋅ d = ((c / d) ⋅ b) ⋅ d | if a = (c / d) ⋅ b, then a ⋅ d = ((c / d) ⋅ b) ⋅ d |
| 5 | ((c / d) ⋅ b) ⋅ d = c ⋅ b | ((c / d) ⋅ b) ⋅ d = c ⋅ b |
| 6 | a ⋅ d = c ⋅ b | if a ⋅ d = ((c / d) ⋅ b) ⋅ d and ((c / d) ⋅ b) ⋅ d = c ⋅ b, then a ⋅ d = c ⋅ b |
| 7 | c ⋅ b = b ⋅ c | c ⋅ b = b ⋅ c |
| 8 | a ⋅ d = b ⋅ c | if a ⋅ d = c ⋅ b and c ⋅ b = b ⋅ c, then a ⋅ d = b ⋅ c |
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