Transitive Property of Equality Variation 1

Add Associative 2

Rearrange Sum Equal 2

Rearrange Sum Equal 3

Transitive Property Application 1

Transitive Property Application 4

Transitive Property Application 5

Transitive Property of Equality Variation 2

Proof: Equation

Let's prove the following theorem:

if b + c = 0, then (a + b) + (c + d) = a + d

Proof:

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Given

1	b + c = 0

Proof Table

#	Claim	Reason
1	(a + b) + (c + d) = (a + (c + d)) + b	(a + b) + (c + d) = (a + (c + d)) + b (add_associative_2)
2	a + (c + d) = (a + c) + d	a + (c + d) = (a + c) + d (associative)
3	(a + (c + d)) + b = ((a + c) + d) + b	if a + (c + d) = (a + c) + d, then (a + (c + d)) + b = ((a + c) + d) + b (Additive Property of Equality)
4	(a + b) + (c + d) = ((a + c) + d) + b	if (a + b) + (c + d) = (a + (c + d)) + b and (a + (c + d)) + b = ((a + c) + d) + b, then (a + b) + (c + d) = ((a + c) + d) + b (Transitive Property of Equality)
5	((a + c) + d) + b = ((b + c) + a) + d	((a + c) + d) + b = ((b + c) + a) + d (equation_a)
6	(b + c) + a = 0 + a	if b + c = 0, then (b + c) + a = 0 + a (Additive Property of Equality)
7	0 + a = a	0 + a = a (zero_plus_a)
8	(b + c) + a = a	if (b + c) + a = 0 + a and 0 + a = a, then (b + c) + a = a (Transitive Property of Equality)
9	((b + c) + a) + d = a + d	if (b + c) + a = a, then ((b + c) + a) + d = a + d (Additive Property of Equality)
10	((a + c) + d) + b = a + d	if ((a + c) + d) + b = ((b + c) + a) + d and ((b + c) + a) + d = a + d, then ((a + c) + d) + b = a + d (Transitive Property of Equality)
11	(a + b) + (c + d) = a + d	if (a + b) + (c + d) = ((a + c) + d) + b and ((a + c) + d) + b = a + d, then (a + b) + (c + d) = a + d (Transitive Property of Equality)

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