Proof: Inequality Problem 2

Let's prove the following theorem:

The inequality properties allow us to divide both sides by 3.

y ⋅ 3 / 3 = y

and

21 / 3 = 7

so

y < 7

Proof:

Proof Table

#	Claim	Reason
1	1 / 3 > 0	1 / 3 > 0 (Fixed Value Statement)
2	(y ⋅ 3) ⋅ (1 / 3) < 21 ⋅ (1 / 3)	if 1 / 3 > 0 and y ⋅ 3 < 21, then (y ⋅ 3) ⋅ (1 / 3) < 21 ⋅ (1 / 3) (Inequality Multiplication Property)
3	21 ⋅ (1 / 3) = 7	21 ⋅ (1 / 3) = 7 (Fixed Value Statement)
4	3 ⋅ (1 / 3) = 1	3 ⋅ (1 / 3) = 1 (Fixed Value Statement)
5	(y ⋅ 3) ⋅ (1 / 3) = y ⋅ (3 ⋅ (1 / 3))	(y ⋅ 3) ⋅ (1 / 3) = y ⋅ (3 ⋅ (1 / 3)) (Associative Property of Multiplication)
6	y ⋅ (3 ⋅ (1 / 3)) < 21 ⋅ (1 / 3)	if (y ⋅ 3) ⋅ (1 / 3) = y ⋅ (3 ⋅ (1 / 3)) and (y ⋅ 3) ⋅ (1 / 3) < 21 ⋅ (1 / 3), then y ⋅ (3 ⋅ (1 / 3)) < 21 ⋅ (1 / 3) (Transitive Property of Inequality 2)
7	y ⋅ (3 ⋅ (1 / 3)) = y ⋅ 1	if 3 ⋅ (1 / 3) = 1, then y ⋅ (3 ⋅ (1 / 3)) = y ⋅ 1 (Multiplicative Property of Equality Variation 1)
8	y ⋅ (3 ⋅ (1 / 3)) = y	if y ⋅ (3 ⋅ (1 / 3)) = y ⋅ 1, then y ⋅ (3 ⋅ (1 / 3)) = y (Multiply by 1)
9	y < 7	if 21 ⋅ (1 / 3) = 7 and y ⋅ (3 ⋅ (1 / 3)) = y and y ⋅ (3 ⋅ (1 / 3)) < 21 ⋅ (1 / 3), then y < 7 (Double Transitive Property Less)

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