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