let s be State of ; :: thesis: for I being InitHalting Program of
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 ; :: 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 )
A1: not a in dom (Initialized I) by SCMFSA6A:48;

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 ( IExec I,s = (Result (s +* (Initialized I))) +* (s | NAT ) & not a in dom (s | NAT ) ) by RELAT_1:86, SCMFSA6B:def 1;
then A2: (IExec I,s) . a = (Result (s +* (Initialized I))) . a by FUNCT_4:12;
ProgramPart (s +* (Initialized I)) halts_on s +* (Initialized I) by Th5, FUNCT_4:26;
then consider n being Element of NAT such that
A3: Result (s +* (Initialized I)) = Computation (s +* (Initialized I)),n and
CurInstr (Result (s +* (Initialized I))) = halt SCM+FSA by AMI_1:def 22;
Initialized I c= s +* (Initialized I) by FUNCT_4:26;
then A4: ( I +* (Start-At (insloc 0 )) c= s +* (Initialized I) & ( for m being Element of NAT st m < n holds
IC (Computation (s +* (Initialized I)),m) in dom I ) ) by Def1, FUNCT_4:26, SCMFSA6B:8;
assume not a in UsedIntLoc I ; :: thesis: (IExec I,s) . a = s . a
hence (IExec I,s) . a = (s +* (Initialized I)) . a by A2, A3, A4, SF_MASTR:69
.= s . a by A1, FUNCT_4:12 ;
:: thesis: verum