Proof: Substitute Add 17

Let's prove the following theorem:

if the following are true:

value of cell 3 at time 18 = (value of cell 3 at time 17) + (value of cell 4 at time 17)
value of cell 3 at time 17 = 2
value of cell 4 at time 17 = 3

then value of cell 3 at time 18 = 5

Proof:

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Given

1	value of cell 3 at time 18 = (value of cell 3 at time 17) + (value of cell 4 at time 17)
2	value of cell 3 at time 17 = 2
3	value of cell 4 at time 17 = 3

Proof Table

#	Claim	Reason
1	(value of cell 3 at time 17) + (value of cell 4 at time 17) = 2 + (value of cell 4 at time 17)	if value of cell 3 at time 17 = 2, then (value of cell 3 at time 17) + (value of cell 4 at time 17) = 2 + (value of cell 4 at time 17) (Substitution)
2	value of cell 3 at time 18 = 2 + (value of cell 4 at time 17)	if value of cell 3 at time 18 = (value of cell 3 at time 17) + (value of cell 4 at time 17) and (value of cell 3 at time 17) + (value of cell 4 at time 17) = 2 + (value of cell 4 at time 17), then value of cell 3 at time 18 = 2 + (value of cell 4 at time 17) (Transitive Property of Equality)
3	2 + (value of cell 4 at time 17) = 2 + 3	if value of cell 4 at time 17 = 3, then 2 + (value of cell 4 at time 17) = 2 + 3 (Substitution)
4	value of cell 3 at time 18 = 2 + 3	if value of cell 3 at time 18 = 2 + (value of cell 4 at time 17) and 2 + (value of cell 4 at time 17) = 2 + 3, then value of cell 3 at time 18 = 2 + 3 (Transitive Property of Equality)
5	2 + 3 = 5	2 + 3 = 5 (Fixed Value Statement)
6	value of cell 3 at time 18 = 5	if value of cell 3 at time 18 = 2 + 3 and 2 + 3 = 5, then value of cell 3 at time 18 = 5 (Transitive Property of Equality)

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