Proof: Algebra Substitution
Let's prove the following theorem:
if the following are true:
- a ⋅ a = (b ⋅ b) + (c ⋅ c)
- a = x
- b = y
then x ⋅ x = (y ⋅ y) + (c ⋅ c)
Proof:
Given
| 1 | a ⋅ a = (b ⋅ b) + (c ⋅ c) |
|---|---|
| 2 | a = x |
| 3 | b = y |
| # | Claim | Reason |
|---|---|---|
| 1 | x ⋅ a = (b ⋅ b) + (c ⋅ c) | if a = x and a ⋅ a = (b ⋅ b) + (c ⋅ c), then x ⋅ a = (b ⋅ b) + (c ⋅ c) |
| 2 | x ⋅ x = (b ⋅ b) + (c ⋅ c) | if x ⋅ a = (b ⋅ b) + (c ⋅ c) and a = x, then x ⋅ x = (b ⋅ b) + (c ⋅ c) |
| 3 | b ⋅ b = y ⋅ b | if b = y, then b ⋅ b = y ⋅ b |
| 4 | y ⋅ b = y ⋅ y | if b = y, then y ⋅ b = y ⋅ y |
| 5 | b ⋅ b = y ⋅ y | if b ⋅ b = y ⋅ b and y ⋅ b = y ⋅ y, then b ⋅ b = y ⋅ y |
| 6 | x ⋅ x = (y ⋅ y) + (c ⋅ c) | if x ⋅ x = (b ⋅ b) + (c ⋅ c) and b ⋅ b = y ⋅ y, then x ⋅ x = (y ⋅ y) + (c ⋅ c) |
Comments
Please log in to add comments