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

for I being really-closed InitHalting Program of SCM+FSA

for a being read-write Int-Location st not a in UsedILoc I holds

(IExec (I,p,s)) . a = s . a

let p be Instruction-Sequence of SCM+FSA; :: thesis: for I being really-closed InitHalting Program of SCM+FSA

for a being read-write Int-Location st not a in UsedILoc I holds

(IExec (I,p,s)) . a = s . a

let I be really-closed InitHalting Program of SCM+FSA; :: thesis: for a being read-write Int-Location st not a in UsedILoc I holds

(IExec (I,p,s)) . a = s . a

let a be read-write Int-Location; :: thesis: ( not a in UsedILoc I implies (IExec (I,p,s)) . a = s . a )

( a <> intloc 0 & a <> IC ) by SCMFSA_2:56;

then A1: not a in dom (Initialize ((intloc 0) .--> 1)) by SCMFSA_M:11, TARSKI:def 2;

A2: (IExec (I,p,s)) . a = (Result ((p +* I),(Initialized s))) . a by SCMFSA6B:def 1;

A3: Initialize ((intloc 0) .--> 1) c= Initialized s by FUNCT_4:25;

A4: I c= p +* I by FUNCT_4:25;

then p +* I halts_on Initialized s by Def1, A3;

then consider n being Nat such that

A5: Result ((p +* I),(Initialized s)) = Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),n) and

CurInstr ((p +* I),(Result ((p +* I),(Initialized s)))) = halt SCM+FSA by EXTPRO_1:def 9;

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

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

then A6: for m being Nat st m < n holds

IC (Comput ((p +* I),(Initialized s),m)) in dom I by A4, AMISTD_1:21;

assume not a in UsedILoc I ; :: thesis: (IExec (I,p,s)) . a = s . a

hence (IExec (I,p,s)) . a = (Initialized s) . a by A2, A5, A6, FUNCT_4:25, SF_MASTR:61

.= s . a by A1, FUNCT_4:11 ;

:: thesis: verum

for I being really-closed InitHalting Program of SCM+FSA

for a being read-write Int-Location st not a in UsedILoc I holds

(IExec (I,p,s)) . a = s . a

let p be Instruction-Sequence of SCM+FSA; :: thesis: for I being really-closed InitHalting Program of SCM+FSA

for a being read-write Int-Location st not a in UsedILoc I holds

(IExec (I,p,s)) . a = s . a

let I be really-closed InitHalting Program of SCM+FSA; :: thesis: for a being read-write Int-Location st not a in UsedILoc I holds

(IExec (I,p,s)) . a = s . a

let a be read-write Int-Location; :: thesis: ( not a in UsedILoc I implies (IExec (I,p,s)) . a = s . a )

( a <> intloc 0 & a <> IC ) by SCMFSA_2:56;

then A1: not a in dom (Initialize ((intloc 0) .--> 1)) by SCMFSA_M:11, TARSKI:def 2;

A2: (IExec (I,p,s)) . a = (Result ((p +* I),(Initialized s))) . a by SCMFSA6B:def 1;

A3: Initialize ((intloc 0) .--> 1) c= Initialized s by FUNCT_4:25;

A4: I c= p +* I by FUNCT_4:25;

then p +* I halts_on Initialized s by Def1, A3;

then consider n being Nat such that

A5: Result ((p +* I),(Initialized s)) = Comput ((p +* I),(s +* (Initialize ((intloc 0) .--> 1))),n) and

CurInstr ((p +* I),(Result ((p +* I),(Initialized s)))) = halt SCM+FSA by EXTPRO_1:def 9;

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

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

then A6: for m being Nat st m < n holds

IC (Comput ((p +* I),(Initialized s),m)) in dom I by A4, AMISTD_1:21;

assume not a in UsedILoc I ; :: thesis: (IExec (I,p,s)) . a = s . a

hence (IExec (I,p,s)) . a = (Initialized s) . a by A2, A5, A6, FUNCT_4:25, SF_MASTR:61

.= s . a by A1, FUNCT_4:11 ;

:: thesis: verum