Proof: Do Control Map At Unchanged 51

Let's prove the following theorem:

if the following are true:

expression state at time 51 = "return"
Control Map at time 51 = [ entry 0: (pair ("while", 2)), [ ] ]

then Control Map at time 52 = [ entry 0: (pair ("while", 2)), [ ] ]

Proof:

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Given

1	expression state at time 51 = "return"
2	Control Map at time 51 = [ entry 0: (pair ("while", 2)), [ ] ]

Proof Table

#	Claim	Reason
1	Control Map at time (51 + 1) = Control Map at time 51	if expression state at time 51 = "return", then Control Map at time (51 + 1) = Control Map at time 51 (Return (10))
2	Control Map at time (51 + 1) = [ entry 0: (pair ("while", 2)), [ ] ]	if Control Map at time (51 + 1) = Control Map at time 51 and Control Map at time 51 = [ entry 0: (pair ("while", 2)), [ ] ], then Control Map at time (51 + 1) = [ entry 0: (pair ("while", 2)), [ ] ] (Transitive Property of Equality)
3	51 + 1 = 52	51 + 1 = 52 (Fixed Value Statement)
4	Control Map at time (51 + 1) = Control Map at time 52	if 51 + 1 = 52, then Control Map at time (51 + 1) = Control Map at time 52 (Substitution)
5	Control Map at time 52 = [ entry 0: (pair ("while", 2)), [ ] ]	if Control Map at time (51 + 1) = Control Map at time 52 and Control Map at time (51 + 1) = [ entry 0: (pair ("while", 2)), [ ] ], then Control Map at time 52 = [ entry 0: (pair ("while", 2)), [ ] ] (Transitive Property of Equality Variation 2)

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