Proof: Subtract Substitute
Let's prove the following theorem:
if a = b, then x - a = x - b
Proof:
Given
| 1 | a = b |
|---|
| # | Claim | Reason |
|---|---|---|
| 1 | x - a = x + (a ⋅ (-1)) | x - a = x + (a ⋅ (-1)) |
| 2 | a ⋅ (-1) = b ⋅ (-1) | if a = b, then a ⋅ (-1) = b ⋅ (-1) |
| 3 | x + (a ⋅ (-1)) = x + (b ⋅ (-1)) | if a ⋅ (-1) = b ⋅ (-1), then x + (a ⋅ (-1)) = x + (b ⋅ (-1)) |
| 4 | x - a = x + (b ⋅ (-1)) | if x - a = x + (a ⋅ (-1)) and x + (a ⋅ (-1)) = x + (b ⋅ (-1)), then x - a = x + (b ⋅ (-1)) |
| 5 | x + (b ⋅ (-1)) = x - b | x + (b ⋅ (-1)) = x - b |
| 6 | x - a = x - b | if x - a = x + (b ⋅ (-1)) and x + (b ⋅ (-1)) = x - b, then x - a = x - b |
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