Proof: Algebra 8

Let's prove the following theorem:

(s / 2) ⋅ (s / 2) = (s ⋅ s) / 4

Proof:

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Proof Table
# 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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