Algebra 1 / Chapter 2: Multiplication / Properties of Multiplication

Proof: Square Product Theorem

Let's prove the following theorem:

if a = b ⋅ c, then a ⋅ a = ((b ⋅ c) ⋅ b) ⋅ c

Proof:

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Given
1 a = b ⋅ c
Proof Table
# Claim Reason
1 a ⋅ a = (b ⋅ c) ⋅ (b ⋅ c) if a = b ⋅ c, then a ⋅ a = (b ⋅ c) ⋅ (b ⋅ c)
2 (b ⋅ c) ⋅ (b ⋅ c) = ((b ⋅ c) ⋅ b) ⋅ c (b ⋅ c) ⋅ (b ⋅ c) = ((b ⋅ c) ⋅ b) ⋅ c
3 a ⋅ a = ((b ⋅ c) ⋅ b) ⋅ c if (b ⋅ c) ⋅ (b ⋅ c) = ((b ⋅ c) ⋅ b) ⋅ c and a ⋅ a = (b ⋅ c) ⋅ (b ⋅ c), then a ⋅ a = ((b ⋅ c) ⋅ b) ⋅ c
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