Proof: Algebra 19
Let's prove the following theorem:
if the following are true:
- a = b + c
- x = y + z
- a = x
- b = z
then c = y
Proof:
Given
| 1 | a = b + c |
|---|---|
| 2 | x = y + z |
| 3 | a = x |
| 4 | b = z |
| # | Claim | Reason |
|---|---|---|
| 1 | a = z + c | if a = b + c and b = z, then a = z + c |
| 2 | x = z + c | if a = x and a = z + c, then x = z + c |
| 3 | x + (z ⋅ (-1)) = c | if x = z + c, then x + (z ⋅ (-1)) = c |
| 4 | x + (z ⋅ (-1)) = y | if x = y + z, then x + (z ⋅ (-1)) = y |
| 5 | c = y | if x + (z ⋅ (-1)) = c and x + (z ⋅ (-1)) = y, then c = y |
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