Proof: Algebra 8
Let's prove the following theorem:
(s / 2) ⋅ (s / 2) = (s ⋅ s) / 4
Proof:
| # | Claim | Reason |
|---|---|---|
| 1 | s / 2 = s ⋅ (1 / 2) | s / 2 = s ⋅ (1 / 2) |
| 2 | (s / 2) ⋅ (s / 2) = (s ⋅ (1 / 2)) ⋅ (s ⋅ (1 / 2)) | if s / 2 = s ⋅ (1 / 2), then (s / 2) ⋅ (s / 2) = (s ⋅ (1 / 2)) ⋅ (s ⋅ (1 / 2)) |
| 3 | (s ⋅ (1 / 2)) ⋅ (s ⋅ (1 / 2)) = ((s ⋅ (1 / 2)) ⋅ s) ⋅ (1 / 2) | (s ⋅ (1 / 2)) ⋅ (s ⋅ (1 / 2)) = ((s ⋅ (1 / 2)) ⋅ s) ⋅ (1 / 2) |
| 4 | ((s ⋅ (1 / 2)) ⋅ s) ⋅ (1 / 2) = ((s ⋅ s) ⋅ (1 / 2)) ⋅ (1 / 2) | ((s ⋅ (1 / 2)) ⋅ s) ⋅ (1 / 2) = ((s ⋅ s) ⋅ (1 / 2)) ⋅ (1 / 2) |
| 5 | (s ⋅ (1 / 2)) ⋅ (s ⋅ (1 / 2)) = ((s ⋅ s) ⋅ (1 / 2)) ⋅ (1 / 2) | if (s ⋅ (1 / 2)) ⋅ (s ⋅ (1 / 2)) = ((s ⋅ (1 / 2)) ⋅ s) ⋅ (1 / 2) and ((s ⋅ (1 / 2)) ⋅ s) ⋅ (1 / 2) = ((s ⋅ s) ⋅ (1 / 2)) ⋅ (1 / 2), then (s ⋅ (1 / 2)) ⋅ (s ⋅ (1 / 2)) = ((s ⋅ s) ⋅ (1 / 2)) ⋅ (1 / 2) |
| 6 | (1 / 2) ⋅ (1 / 2) = 1 / 4 | (1 / 2) ⋅ (1 / 2) = 1 / 4 |
| 7 | ((s ⋅ s) ⋅ (1 / 2)) ⋅ (1 / 2) = (s ⋅ s) ⋅ ((1 / 2) ⋅ (1 / 2)) | ((s ⋅ s) ⋅ (1 / 2)) ⋅ (1 / 2) = (s ⋅ s) ⋅ ((1 / 2) ⋅ (1 / 2)) |
| 8 | ((s ⋅ s) ⋅ (1 / 2)) ⋅ (1 / 2) = (s ⋅ s) ⋅ (1 / 4) | if ((s ⋅ s) ⋅ (1 / 2)) ⋅ (1 / 2) = (s ⋅ s) ⋅ ((1 / 2) ⋅ (1 / 2)) and (1 / 2) ⋅ (1 / 2) = 1 / 4, then ((s ⋅ s) ⋅ (1 / 2)) ⋅ (1 / 2) = (s ⋅ s) ⋅ (1 / 4) |
| 9 | (s ⋅ s) / 4 = (s ⋅ s) ⋅ (1 / 4) | (s ⋅ s) / 4 = (s ⋅ s) ⋅ (1 / 4) |
| 10 | ((s ⋅ s) ⋅ (1 / 2)) ⋅ (1 / 2) = (s ⋅ s) / 4 | if ((s ⋅ s) ⋅ (1 / 2)) ⋅ (1 / 2) = (s ⋅ s) ⋅ (1 / 4) and (s ⋅ s) / 4 = (s ⋅ s) ⋅ (1 / 4), then ((s ⋅ s) ⋅ (1 / 2)) ⋅ (1 / 2) = (s ⋅ s) / 4 |
| 11 | (s ⋅ (1 / 2)) ⋅ (s ⋅ (1 / 2)) = (s ⋅ s) / 4 | if (s ⋅ (1 / 2)) ⋅ (s ⋅ (1 / 2)) = ((s ⋅ s) ⋅ (1 / 2)) ⋅ (1 / 2) and ((s ⋅ s) ⋅ (1 / 2)) ⋅ (1 / 2) = (s ⋅ s) / 4, then (s ⋅ (1 / 2)) ⋅ (s ⋅ (1 / 2)) = (s ⋅ s) / 4 |
| 12 | (s / 2) ⋅ (s / 2) = (s ⋅ s) / 4 | if (s / 2) ⋅ (s / 2) = (s ⋅ (1 / 2)) ⋅ (s ⋅ (1 / 2)) and (s ⋅ (1 / 2)) ⋅ (s ⋅ (1 / 2)) = (s ⋅ s) / 4, then (s / 2) ⋅ (s / 2) = (s ⋅ s) / 4 |
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