let s be State of SCM+FSA ; :: thesis: for I being InitHalting Program of SCM+FSA
for a being read-write Int-Location st not a in UsedIntLoc I holds
(IExec I,s) . a = s . a
let I be InitHalting Program of SCM+FSA ; :: thesis: for a being read-write Int-Location st not a in UsedIntLoc I holds
(IExec I,s) . a = s . a
let a be read-write Int-Location ; :: thesis: ( not a in UsedIntLoc I implies (IExec I,s) . a = s . a )
assume A1: not a in UsedIntLoc I ; :: thesis: (IExec I,s) . a = s . a
A2: IExec I,s = (Result (s +* (Initialized I))) +* (s | NAT ) by SCMFSA6B:def 1;

now

assume a in NAT ; :: thesis: contradiction
then reconsider a = a as Instruction-Location of SCM+FSA by AMI_1:def 4;
a = a ;
hence contradiction by SCMFSA_2:84; :: thesis: verum

end;

then not a in dom (s | NAT ) by RELAT_1:86;
then A3: (IExec I,s) . a = (Result (s +* (Initialized I))) . a by A2, FUNCT_4:12;
A4: Initialized I c= s +* (Initialized I) by FUNCT_4:26;
s +* (Initialized I) is halting by Th5, FUNCT_4:26;
then consider n being Element of NAT such that
A5: ( Result (s +* (Initialized I)) = Computation (s +* (Initialized I)),n & CurInstr (Result (s +* (Initialized I))) = halt SCM+FSA ) by AMI_1:def 22;
A6: I +* (Start-At (insloc 0 )) c= s +* (Initialized I) by FUNCT_4:26, SCMFSA6B:8;
A7: for m being Element of NAT st m < n holds
IC (Computation (s +* (Initialized I)),m) in dom I by A4, Def1;
A8: ( not a in dom (Initialized I) & a in dom s ) by SCMFSA6A:48, SCMFSA_2:66;
thus (IExec I,s) . a = (s +* (Initialized I)) . a by A1, A3, A5, A6, A7, SF_MASTR:69
.= s . a by A8, FUNCT_4:12 ; :: thesis: verum