let s be State of SCM+FSA; :: thesis: for p being Instruction-Sequence of SCM+FSA

for b being Int-Location

for I being really-closed Program of st I is_halting_on Initialized s,p & not b in UsedILoc I holds

(IExec (I,p,s)) . b = (Initialized s) . b

let p be Instruction-Sequence of SCM+FSA; :: thesis: for b being Int-Location

for I being really-closed Program of st I is_halting_on Initialized s,p & not b in UsedILoc I holds

(IExec (I,p,s)) . b = (Initialized s) . b

let b be Int-Location; :: thesis: for I being really-closed Program of st I is_halting_on Initialized s,p & not b in UsedILoc I holds

(IExec (I,p,s)) . b = (Initialized s) . b

let I be really-closed Program of ; :: thesis: ( I is_halting_on Initialized s,p & not b in UsedILoc I implies (IExec (I,p,s)) . b = (Initialized s) . b )

set a = b;

assume that

A1: I is_halting_on Initialized s,p and

A2: not b in UsedILoc I ; :: thesis: (IExec (I,p,s)) . b = (Initialized s) . b

set Is = Initialized s;

set pI = p +* I;

A3: p +* I halts_on Initialize (Initialized s) by A1, SCMFSA7B:def 7;

A4: Initialized s = Initialize (Initialized s) by MEMSTR_0:44;

A5: I c= p +* I by FUNCT_4:25;

IC (Initialized s) = 0 by MEMSTR_0:def 11;

then IC (Initialized s) in dom I by AFINSQ_1:65;

then for m being Nat st m < LifeSpan ((p +* I),(Initialized s)) holds

IC (Comput ((p +* I),(Initialized s),m)) in dom I by A5, AMISTD_1:21;

then A6: (Comput ((p +* I),(Initialized s),(LifeSpan ((p +* I),(Initialized s))))) . b = (Initialized s) . b by A2, FUNCT_4:25, SF_MASTR:61;

DataPart (IExec (I,p,s)) = DataPart (Result ((p +* I),(Initialized s))) by SCMFSA6B:def 1

.= DataPart (Result ((p +* I),(Initialized s)))

.= DataPart (Comput ((p +* I),(Initialized s),(LifeSpan ((p +* I),(Initialized s))))) by A4, A3, EXTPRO_1:23 ;

hence (IExec (I,p,s)) . b = (Initialized s) . b by A6, SCMFSA_M:2; :: thesis: verum

for b being Int-Location

for I being really-closed Program of st I is_halting_on Initialized s,p & not b in UsedILoc I holds

(IExec (I,p,s)) . b = (Initialized s) . b

let p be Instruction-Sequence of SCM+FSA; :: thesis: for b being Int-Location

for I being really-closed Program of st I is_halting_on Initialized s,p & not b in UsedILoc I holds

(IExec (I,p,s)) . b = (Initialized s) . b

let b be Int-Location; :: thesis: for I being really-closed Program of st I is_halting_on Initialized s,p & not b in UsedILoc I holds

(IExec (I,p,s)) . b = (Initialized s) . b

let I be really-closed Program of ; :: thesis: ( I is_halting_on Initialized s,p & not b in UsedILoc I implies (IExec (I,p,s)) . b = (Initialized s) . b )

set a = b;

assume that

A1: I is_halting_on Initialized s,p and

A2: not b in UsedILoc I ; :: thesis: (IExec (I,p,s)) . b = (Initialized s) . b

set Is = Initialized s;

set pI = p +* I;

A3: p +* I halts_on Initialize (Initialized s) by A1, SCMFSA7B:def 7;

A4: Initialized s = Initialize (Initialized s) by MEMSTR_0:44;

A5: I c= p +* I by FUNCT_4:25;

IC (Initialized s) = 0 by MEMSTR_0:def 11;

then IC (Initialized s) in dom I by AFINSQ_1:65;

then for m being Nat st m < LifeSpan ((p +* I),(Initialized s)) holds

IC (Comput ((p +* I),(Initialized s),m)) in dom I by A5, AMISTD_1:21;

then A6: (Comput ((p +* I),(Initialized s),(LifeSpan ((p +* I),(Initialized s))))) . b = (Initialized s) . b by A2, FUNCT_4:25, SF_MASTR:61;

DataPart (IExec (I,p,s)) = DataPart (Result ((p +* I),(Initialized s))) by SCMFSA6B:def 1

.= DataPart (Result ((p +* I),(Initialized s)))

.= DataPart (Comput ((p +* I),(Initialized s),(LifeSpan ((p +* I),(Initialized s))))) by A4, A3, EXTPRO_1:23 ;

hence (IExec (I,p,s)) . b = (Initialized s) . b by A6, SCMFSA_M:2; :: thesis: verum