Proof: Subtract From Both Sides

Let's prove the following theorem:

if a = b, then b - c = a - c

Proof:

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

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