let s be State of SCM+FSA ; :: thesis: for I being InitHalting Program of SCM+FSA
for f being FinSeq-Location st not f in UsedInt*Loc I holds
(IExec I,s) . f = s . f
let I be InitHalting Program of SCM+FSA ; :: thesis: for f being FinSeq-Location st not f in UsedInt*Loc I holds
(IExec I,s) . f = s . f
let f be FinSeq-Location ; :: thesis: ( not f in UsedInt*Loc I implies (IExec I,s) . f = s . f )
A1: not f in dom (Initialized I) by SCMFSA6A:49;
not f in NAT by SCMFSA_2:85;
then ( IExec I,s = (Result (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) +* (s | NAT ) & not f in dom (s | NAT ) ) by RELAT_1:86, SCMFSA6B:def 1;
then A2: (IExec I,s) . f = (Result (ProgramPart (s +* (Initialized I))),(s +* (Initialized I))) . f 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 (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)) = Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),n and
CurInstr (ProgramPart (s +* (Initialized I))),(Result (ProgramPart (s +* (Initialized I))),(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 0 ,SCM+FSA ) c= s +* (Initialized I) & ( for m being Element of NAT st m < n holds
IC (Comput (ProgramPart (s +* (Initialized I))),(s +* (Initialized I)),m) in dom I ) ) by Def1, SCMFSA6B:8;
assume not f in UsedInt*Loc I ; :: thesis: (IExec I,s) . f = s . f
hence (IExec I,s) . f = (s +* (Initialized I)) . f by A2, A3, A4, SF_MASTR:71
.= s . f by A1, FUNCT_4:12 ;
:: thesis: verum