Let's prove the following theorem:

if the following are true:

then c = a / b

Given

1	a = b ⋅ c
2	not (b = 0)

Proof Table

#	Claim	Reason
1	(1 / b) ⋅ a = (1 / b) ⋅ (b ⋅ c)	if a = b ⋅ c, then (1 / b) ⋅ a = (1 / b) ⋅ (b ⋅ c) (Substitution)
2	(1 / b) ⋅ (b ⋅ c) = c	if not (b = 0), then (1 / b) ⋅ (b ⋅ c) = c (Remove One 2)
3	(1 / b) ⋅ a = c	if (1 / b) ⋅ (b ⋅ c) = c and (1 / b) ⋅ a = (1 / b) ⋅ (b ⋅ c), then (1 / b) ⋅ a = c (Transitive Property of Equality)
4	(1 / b) ⋅ a = a / b	(1 / b) ⋅ a = a / b (multiply_by_inverse_2)
5	a / b = c	if (1 / b) ⋅ a = c and (1 / b) ⋅ a = a / b, then a / b = c (Transitive Property of Equality Variation 2)
6	c = a / b	if a / b = c, then c = a / b (Symmetric Property of Equality)

Previous Lesson Next Lesson

Please log in to add comments