Proof: Product is One 3

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 ⋅ b) ⋅ (1 / a)) ⋅ (1 / b) = 1 if not (b = 0) and not (a = 0), then ((a ⋅ b) ⋅ (1 / a)) ⋅ (1 / b) = 1
2 ((a ⋅ b) ⋅ (1 / a)) ⋅ (1 / b) = (a ⋅ b) ⋅ ((1 / a) ⋅ (1 / b)) ((a ⋅ b) ⋅ (1 / a)) ⋅ (1 / b) = (a ⋅ b) ⋅ ((1 / a) ⋅ (1 / b))
3 (a ⋅ b) ⋅ ((1 / a) ⋅ (1 / b)) = 1 if ((a ⋅ b) ⋅ (1 / a)) ⋅ (1 / b) = 1 and ((a ⋅ b) ⋅ (1 / a)) ⋅ (1 / b) = (a ⋅ b) ⋅ ((1 / a) ⋅ (1 / b)), then (a ⋅ b) ⋅ ((1 / a) ⋅ (1 / b)) = 1

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