Proof: Cross Multiply Theorem

Let's prove the following theorem:

if a / b = c / d, then a ⋅ d = b ⋅ c

Proof:

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Given
1 a / b = c / d
Proof Table
# Claim Reason
1 (a / b) ⋅ b = (c / d) ⋅ b if a / b = c / d, then (a / b) ⋅ b = (c / d) ⋅ b
2 (a / b) ⋅ b = a (a / b) ⋅ b = a
3 a = (c / d) ⋅ b if (a / b) ⋅ b = a and (a / b) ⋅ b = (c / d) ⋅ b, then a = (c / d) ⋅ b
4 a ⋅ d = ((c / d) ⋅ b) ⋅ d if a = (c / d) ⋅ b, then a ⋅ d = ((c / d) ⋅ b) ⋅ d
5 ((c / d) ⋅ b) ⋅ d = c ⋅ b ((c / d) ⋅ b) ⋅ d = c ⋅ b
6 a ⋅ d = c ⋅ b if a ⋅ d = ((c / d) ⋅ b) ⋅ d and ((c / d) ⋅ b) ⋅ d = c ⋅ b, then a ⋅ d = c ⋅ b
7 c ⋅ b = b ⋅ c c ⋅ b = b ⋅ c
8 a ⋅ d = b ⋅ c if a ⋅ d = c ⋅ b and c ⋅ b = b ⋅ c, then a ⋅ d = b ⋅ c

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