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:
Given
| 1 | a = b |
|---|---|
| 2 | c = d |
| # | 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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