Proof: Divide Each Side

Let's prove the following theorem:

if a ⋅ b = c, then a = c / b

Proof:

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Given
1 a ⋅ b = c
Proof Table
# 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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