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:

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Given

1	a ⋅ a = (b ⋅ b) + (c ⋅ c)
2	a = x
3	b = y

Proof Table

#	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) (Substitution 14)
2	x ⋅ x = (b ⋅ b) + (c ⋅ c)	if x ⋅ a = (b ⋅ b) + (c ⋅ c) and a = x, then x ⋅ x = (b ⋅ b) + (c ⋅ c) (Substitution in Product)
3	b ⋅ b = y ⋅ b	if b = y, then b ⋅ b = y ⋅ b (Multiplicative Property of Equality)
4	y ⋅ b = y ⋅ y	if b = y, then y ⋅ b = y ⋅ y (Multiplicative Property of Equality Variation 1)
5	b ⋅ b = y ⋅ y	if b ⋅ b = y ⋅ b and y ⋅ b = y ⋅ y, then b ⋅ b = y ⋅ y (Transitive Property of Equality)
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) (Term 1 Substitution)

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