Proof: Multiply by One

Let's prove the following theorem:

if not (c = 0), then (a ⋅ (1 / c)) ⋅ c = a

Proof:

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Given
1 not (c = 0)
Proof Table
# Claim Reason
1 (1 / c) ⋅ c = 1 if not (c = 0), then (1 / c) ⋅ c = 1
2 (a ⋅ (1 / c)) ⋅ c = a ⋅ ((1 / c) ⋅ c) (a ⋅ (1 / c)) ⋅ c = a ⋅ ((1 / c) ⋅ c)
3 a ⋅ ((1 / c) ⋅ c) = a ⋅ 1 if (1 / c) ⋅ c = 1, then a ⋅ ((1 / c) ⋅ c) = a ⋅ 1
4 a ⋅ 1 = a a ⋅ 1 = a
5 a ⋅ ((1 / c) ⋅ c) = a if a ⋅ 1 = a and a ⋅ ((1 / c) ⋅ c) = a ⋅ 1, then a ⋅ ((1 / c) ⋅ c) = a
6 (a ⋅ (1 / c)) ⋅ c = a if a ⋅ ((1 / c) ⋅ c) = a and (a ⋅ (1 / c)) ⋅ c = a ⋅ ((1 / c) ⋅ c), then (a ⋅ (1 / c)) ⋅ c = a

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