Proof: Division is Commutative
Let's prove the following theorem:
a ⋅ (b / c) = (a ⋅ b) / c
Proof:
| # | Claim | Reason |
|---|---|---|
| 1 | b / c = b ⋅ (1 / c) | b / c = b ⋅ (1 / c) |
| 2 | (a ⋅ b) / c = (a ⋅ b) ⋅ (1 / c) | (a ⋅ b) / c = (a ⋅ b) ⋅ (1 / c) |
| 3 | (a ⋅ b) ⋅ (1 / c) = a ⋅ (b ⋅ (1 / c)) | (a ⋅ b) ⋅ (1 / c) = a ⋅ (b ⋅ (1 / c)) |
| 4 | a ⋅ (b ⋅ (1 / c)) = a ⋅ (b / c) | if b / c = b ⋅ (1 / c), then a ⋅ (b ⋅ (1 / c)) = a ⋅ (b / c) |
| 5 | (a ⋅ b) ⋅ (1 / c) = a ⋅ (b / c) | if (a ⋅ b) ⋅ (1 / c) = a ⋅ (b ⋅ (1 / c)) and a ⋅ (b ⋅ (1 / c)) = a ⋅ (b / c), then (a ⋅ b) ⋅ (1 / c) = a ⋅ (b / c) |
| 6 | a ⋅ (b / c) = (a ⋅ b) / c | if (a ⋅ b) / c = (a ⋅ b) ⋅ (1 / c) and (a ⋅ b) ⋅ (1 / c) = a ⋅ (b / c), then a ⋅ (b / c) = (a ⋅ b) / c |
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