Proof: Substitute First Term
Let's prove the following theorem:
if the following are true:
- (a + b) + c = d
- a = e
then (e + b) + c = d
Proof:
Given
| 1 | (a + b) + c = d |
|---|---|
| 2 | a = e |
| # | Claim | Reason |
|---|---|---|
| 1 | (a + b) + c = a + (b + c) | (a + b) + c = a + (b + c) |
| 2 | (b + c) + a = a + (b + c) | (b + c) + a = a + (b + c) |
| 3 | a + (b + c) = (a + b) + c | if (a + b) + c = a + (b + c), then a + (b + c) = (a + b) + c |
| 4 | (b + c) + a = (a + b) + c | if (b + c) + a = a + (b + c) and a + (b + c) = (a + b) + c, then (b + c) + a = (a + b) + c |
| 5 | (b + c) + a = d | if (b + c) + a = (a + b) + c and (a + b) + c = d, then (b + c) + a = d |
| 6 | (b + c) + e = d | if (b + c) + a = d and a = e, then (b + c) + e = d |
| 7 | e + (b + c) = d | if (b + c) + e = d, then e + (b + c) = d |
| 8 | (e + b) + c = e + (b + c) | (e + b) + c = e + (b + c) |
| 9 | (e + b) + c = d | if (e + b) + c = e + (b + c) and e + (b + c) = d, then (e + b) + c = d |
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