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