Proof: Substitute Add 10

Let's prove the following theorem:

if the following are true:

value of cell 3 at time 11 = (value of cell 1 at time 10) + (value of cell 2 at time 10)
value of cell 1 at time 10 = 1
value of cell 2 at time 10 = 1

then value of cell 3 at time 11 = 2

Proof:

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Given

1	value of cell 3 at time 11 = (value of cell 1 at time 10) + (value of cell 2 at time 10)
2	value of cell 1 at time 10 = 1
3	value of cell 2 at time 10 = 1

Proof Table

#	Claim	Reason
1	(value of cell 1 at time 10) + (value of cell 2 at time 10) = 1 + (value of cell 2 at time 10)	if value of cell 1 at time 10 = 1, then (value of cell 1 at time 10) + (value of cell 2 at time 10) = 1 + (value of cell 2 at time 10) (Substitution)
2	value of cell 3 at time 11 = 1 + (value of cell 2 at time 10)	if value of cell 3 at time 11 = (value of cell 1 at time 10) + (value of cell 2 at time 10) and (value of cell 1 at time 10) + (value of cell 2 at time 10) = 1 + (value of cell 2 at time 10), then value of cell 3 at time 11 = 1 + (value of cell 2 at time 10) (Transitive Property of Equality)
3	1 + (value of cell 2 at time 10) = 1 + 1	if value of cell 2 at time 10 = 1, then 1 + (value of cell 2 at time 10) = 1 + 1 (Substitution)
4	value of cell 3 at time 11 = 1 + 1	if value of cell 3 at time 11 = 1 + (value of cell 2 at time 10) and 1 + (value of cell 2 at time 10) = 1 + 1, then value of cell 3 at time 11 = 1 + 1 (Transitive Property of Equality)
5	1 + 1 = 2	1 + 1 = 2 (Fixed Value Statement)
6	value of cell 3 at time 11 = 2	if value of cell 3 at time 11 = 1 + 1 and 1 + 1 = 2, then value of cell 3 at time 11 = 2 (Transitive Property of Equality)

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