Proof: Square

Let's prove the following theorem:

Proof:

Proof Table

#	Claim	Reason
1	((((a ⋅ b) ⋅ 2) + (a ⋅ a)) + ((a ⋅ b) ⋅ (-2))) + (b ⋅ b) = ((((a ⋅ b) ⋅ 2) + ((a ⋅ b) ⋅ (-2))) + (a ⋅ a)) + (b ⋅ b)	((((a ⋅ b) ⋅ 2) + (a ⋅ a)) + ((a ⋅ b) ⋅ (-2))) + (b ⋅ b) = ((((a ⋅ b) ⋅ 2) + ((a ⋅ b) ⋅ (-2))) + (a ⋅ a)) + (b ⋅ b) (swap_b_and_c)
2	((a ⋅ b) ⋅ 2) + ((a ⋅ b) ⋅ (-2)) = 0	((a ⋅ b) ⋅ 2) + ((a ⋅ b) ⋅ (-2)) = 0 (additive_inverse_2)
3	((((a ⋅ b) ⋅ 2) + ((a ⋅ b) ⋅ (-2))) + (a ⋅ a)) + (b ⋅ b) = (((a ⋅ b) ⋅ 2) + ((a ⋅ b) ⋅ (-2))) + ((a ⋅ a) + (b ⋅ b))	((((a ⋅ b) ⋅ 2) + ((a ⋅ b) ⋅ (-2))) + (a ⋅ a)) + (b ⋅ b) = (((a ⋅ b) ⋅ 2) + ((a ⋅ b) ⋅ (-2))) + ((a ⋅ a) + (b ⋅ b)) (Associative Property of Addition)
4	((((a ⋅ b) ⋅ 2) + ((a ⋅ b) ⋅ (-2))) + (a ⋅ a)) + (b ⋅ b) = 0 + ((a ⋅ a) + (b ⋅ b))	if ((((a ⋅ b) ⋅ 2) + ((a ⋅ b) ⋅ (-2))) + (a ⋅ a)) + (b ⋅ b) = (((a ⋅ b) ⋅ 2) + ((a ⋅ b) ⋅ (-2))) + ((a ⋅ a) + (b ⋅ b)) and ((a ⋅ b) ⋅ 2) + ((a ⋅ b) ⋅ (-2)) = 0, then ((((a ⋅ b) ⋅ 2) + ((a ⋅ b) ⋅ (-2))) + (a ⋅ a)) + (b ⋅ b) = 0 + ((a ⋅ a) + (b ⋅ b)) (Term 1 Substitution)
5	0 + ((a ⋅ a) + (b ⋅ b)) = (a ⋅ a) + (b ⋅ b)	0 + ((a ⋅ a) + (b ⋅ b)) = (a ⋅ a) + (b ⋅ b) (Additive Identity Variation)
6	((((a ⋅ b) ⋅ 2) + ((a ⋅ b) ⋅ (-2))) + (a ⋅ a)) + (b ⋅ b) = (a ⋅ a) + (b ⋅ b)	if ((((a ⋅ b) ⋅ 2) + ((a ⋅ b) ⋅ (-2))) + (a ⋅ a)) + (b ⋅ b) = 0 + ((a ⋅ a) + (b ⋅ b)) and 0 + ((a ⋅ a) + (b ⋅ b)) = (a ⋅ a) + (b ⋅ b), then ((((a ⋅ b) ⋅ 2) + ((a ⋅ b) ⋅ (-2))) + (a ⋅ a)) + (b ⋅ b) = (a ⋅ a) + (b ⋅ b) (Transitive Property of Equality)
7	((((a ⋅ b) ⋅ 2) + (a ⋅ a)) + ((a ⋅ b) ⋅ (-2))) + (b ⋅ b) = ((a ⋅ b) ⋅ 2) + (((a ⋅ a) + ((a ⋅ b) ⋅ (-2))) + (b ⋅ b))	((((a ⋅ b) ⋅ 2) + (a ⋅ a)) + ((a ⋅ b) ⋅ (-2))) + (b ⋅ b) = ((a ⋅ b) ⋅ 2) + (((a ⋅ a) + ((a ⋅ b) ⋅ (-2))) + (b ⋅ b)) (add_associative)
8	((a ⋅ b) ⋅ 2) + (((a ⋅ a) + ((a ⋅ b) ⋅ (-2))) + (b ⋅ b)) = ((((a ⋅ b) ⋅ 2) + (a ⋅ a)) + ((a ⋅ b) ⋅ (-2))) + (b ⋅ b)	if ((((a ⋅ b) ⋅ 2) + (a ⋅ a)) + ((a ⋅ b) ⋅ (-2))) + (b ⋅ b) = ((a ⋅ b) ⋅ 2) + (((a ⋅ a) + ((a ⋅ b) ⋅ (-2))) + (b ⋅ b)), then ((a ⋅ b) ⋅ 2) + (((a ⋅ a) + ((a ⋅ b) ⋅ (-2))) + (b ⋅ b)) = ((((a ⋅ b) ⋅ 2) + (a ⋅ a)) + ((a ⋅ b) ⋅ (-2))) + (b ⋅ b) (Symmetric Property of Equality)
9	((((a ⋅ b) ⋅ 2) + (a ⋅ a)) + ((a ⋅ b) ⋅ (-2))) + (b ⋅ b) = (a ⋅ a) + (b ⋅ b)	if ((((a ⋅ b) ⋅ 2) + (a ⋅ a)) + ((a ⋅ b) ⋅ (-2))) + (b ⋅ b) = ((((a ⋅ b) ⋅ 2) + ((a ⋅ b) ⋅ (-2))) + (a ⋅ a)) + (b ⋅ b) and ((((a ⋅ b) ⋅ 2) + ((a ⋅ b) ⋅ (-2))) + (a ⋅ a)) + (b ⋅ b) = (a ⋅ a) + (b ⋅ b), then ((((a ⋅ b) ⋅ 2) + (a ⋅ a)) + ((a ⋅ b) ⋅ (-2))) + (b ⋅ b) = (a ⋅ a) + (b ⋅ b) (Transitive Property of Equality)
10	((a ⋅ b) ⋅ 2) + (((a ⋅ a) + ((a ⋅ b) ⋅ (-2))) + (b ⋅ b)) = (a ⋅ a) + (b ⋅ b)	if ((a ⋅ b) ⋅ 2) + (((a ⋅ a) + ((a ⋅ b) ⋅ (-2))) + (b ⋅ b)) = ((((a ⋅ b) ⋅ 2) + (a ⋅ a)) + ((a ⋅ b) ⋅ (-2))) + (b ⋅ b) and ((((a ⋅ b) ⋅ 2) + (a ⋅ a)) + ((a ⋅ b) ⋅ (-2))) + (b ⋅ b) = (a ⋅ a) + (b ⋅ b), then ((a ⋅ b) ⋅ 2) + (((a ⋅ a) + ((a ⋅ b) ⋅ (-2))) + (b ⋅ b)) = (a ⋅ a) + (b ⋅ b) (Transitive Property of Equality)

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