Proof: Divide Numerators

Let's prove the following theorem:

if the following are true:
  • a / c = b / c
  • not (c = 0)

then a = b

Proof:

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

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