Proof: Subtract Same Sides

Let's prove the following theorem:

if the following are true:
  • a = b
  • c = d

then a - c = b - d

Proof:

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

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