Proof: Product is One 2

Let's prove the following theorem:

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

then ((a ⋅ b) ⋅ (1 / a)) ⋅ (1 / b) = 1

Proof:

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

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