Proof: Substitute Add 16

Let's prove the following theorem:

if the following are true:

value of cell 3 at time 17 = (value of cell 1 at time 16) + (value of cell 2 at time 16)
value of cell 1 at time 16 = 1
value of cell 2 at time 16 = 2

then value of cell 3 at time 17 = 3

Proof:

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Given

1	value of cell 3 at time 17 = (value of cell 1 at time 16) + (value of cell 2 at time 16)
2	value of cell 1 at time 16 = 1
3	value of cell 2 at time 16 = 2

Proof Table

#	Claim	Reason
1	(value of cell 1 at time 16) + (value of cell 2 at time 16) = 1 + (value of cell 2 at time 16)	if value of cell 1 at time 16 = 1, then (value of cell 1 at time 16) + (value of cell 2 at time 16) = 1 + (value of cell 2 at time 16) (Substitution)
2	value of cell 3 at time 17 = 1 + (value of cell 2 at time 16)	if value of cell 3 at time 17 = (value of cell 1 at time 16) + (value of cell 2 at time 16) and (value of cell 1 at time 16) + (value of cell 2 at time 16) = 1 + (value of cell 2 at time 16), then value of cell 3 at time 17 = 1 + (value of cell 2 at time 16) (Transitive Property of Equality)
3	1 + (value of cell 2 at time 16) = 1 + 2	if value of cell 2 at time 16 = 2, then 1 + (value of cell 2 at time 16) = 1 + 2 (Substitution)
4	value of cell 3 at time 17 = 1 + 2	if value of cell 3 at time 17 = 1 + (value of cell 2 at time 16) and 1 + (value of cell 2 at time 16) = 1 + 2, then value of cell 3 at time 17 = 1 + 2 (Transitive Property of Equality)
5	1 + 2 = 3	1 + 2 = 3 (Fixed Value Statement)
6	value of cell 3 at time 17 = 3	if value of cell 3 at time 17 = 1 + 2 and 1 + 2 = 3, then value of cell 3 at time 17 = 3 (Transitive Property of Equality)

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