Proof: Product is One

Let's prove the following theorem:

if the following are true:

not (a = 0)
not (b = 0)

then (a ⋅ (1 / a)) ⋅ (b ⋅ (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) = 1	if not (a = 0), then a ⋅ (1 / a) = 1 (Algebra 17)
2	b ⋅ (1 / b) = 1	if not (b = 0), then b ⋅ (1 / b) = 1 (Algebra 17)
3	(a ⋅ (1 / a)) ⋅ (b ⋅ (1 / b)) = 1 ⋅ (b ⋅ (1 / b))	if a ⋅ (1 / a) = 1, then (a ⋅ (1 / a)) ⋅ (b ⋅ (1 / b)) = 1 ⋅ (b ⋅ (1 / b)) (Substitution)
4	1 ⋅ (b ⋅ (1 / b)) = 1 ⋅ 1	if b ⋅ (1 / b) = 1, then 1 ⋅ (b ⋅ (1 / b)) = 1 ⋅ 1 (Substitution)
5	1 ⋅ 1 = 1	1 ⋅ 1 = 1 (Multiplicative Identity)
6	1 ⋅ (b ⋅ (1 / b)) = 1	if 1 ⋅ (b ⋅ (1 / b)) = 1 ⋅ 1 and 1 ⋅ 1 = 1, then 1 ⋅ (b ⋅ (1 / b)) = 1 (Transitive Property of Equality)
7	(a ⋅ (1 / a)) ⋅ (b ⋅ (1 / b)) = 1	if (a ⋅ (1 / a)) ⋅ (b ⋅ (1 / b)) = 1 ⋅ (b ⋅ (1 / b)) and 1 ⋅ (b ⋅ (1 / b)) = 1, then (a ⋅ (1 / a)) ⋅ (b ⋅ (1 / b)) = 1 (Transitive Property of Equality)

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