Proof: Subtract Substitute 2 Vars

Let's prove the following theorem:

if the following are true:

a = b
c = d

then a - c = b - d

Proof:

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Given

1	a = b
2	c = d

Proof Table

#	Claim	Reason
1	a + (c ⋅ (-1)) = b + (c ⋅ (-1))	if a = b, then a + (c ⋅ (-1)) = b + (c ⋅ (-1)) (Additive Property of Equality)
2	c ⋅ (-1) = d ⋅ (-1)	if c = d, then c ⋅ (-1) = d ⋅ (-1) (Multiplicative Property of Equality)
3	a + (c ⋅ (-1)) = b + (d ⋅ (-1))	if c ⋅ (-1) = d ⋅ (-1) and a + (c ⋅ (-1)) = b + (c ⋅ (-1)), then a + (c ⋅ (-1)) = b + (d ⋅ (-1)) (Substitution 2)
4	a + (c ⋅ (-1)) = a - c	a + (c ⋅ (-1)) = a - c (subtraction_example)
5	b + (d ⋅ (-1)) = b - d	b + (d ⋅ (-1)) = b - d (subtraction_example)
6	a - c = b - d	if b + (d ⋅ (-1)) = b - d and a + (c ⋅ (-1)) = a - c and a + (c ⋅ (-1)) = b + (d ⋅ (-1)), then a - c = b - d (Transitive Property Application 2)

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