Proof: Distribute Subtract2

Let's prove the following theorem:

(c ⋅ a) - (c ⋅ b) = c ⋅ (a - b)

Proof:

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

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