let f be FinSeq-Location ; :: thesis: for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA
for I being really-closed parahalting Program of st not f in UsedI*Loc I holds
(IExec (I,P,s)) . f = s . f
let s be State of SCM+FSA; :: thesis: for P being Instruction-Sequence of SCM+FSA
for I being really-closed parahalting Program of st not f in UsedI*Loc I holds
(IExec (I,P,s)) . f = s . f
let P be Instruction-Sequence of SCM+FSA; :: thesis: for I being really-closed parahalting Program of st not f in UsedI*Loc I holds
(IExec (I,P,s)) . f = s . f
let I be really-closed parahalting Program of ; :: thesis: ( not f in UsedI*Loc I implies (IExec (I,P,s)) . f = s . f )
assume A1: not f in UsedI*Loc I ; :: thesis: (IExec (I,P,s)) . f = s . f
A2: I c= P +* I by FUNCT_4:25;
A3: Initialize ((intloc 0) .--> 1) c= s +* (Initialize ((intloc 0) .--> 1)) by FUNCT_4:25;
then P +* I halts_on s +* (Initialize ((intloc 0) .--> 1)) by Th2, A2;
then consider n being Nat such that
A4: Result ((P +* I),(s +* (Initialize ((intloc 0) .--> 1)))) = Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),n) and
CurInstr ((P +* I),(Result ((P +* I),(s +* (Initialize ((intloc 0) .--> 1)))))) = halt SCM+FSA by EXTPRO_1:def 9;
A5: dom (Initialize ((intloc 0) .--> 1)) = (dom ((intloc 0) .--> 1)) \/ (dom (Start-At (0,SCM+FSA))) by FUNCT_4:def 1;
A6: not f in dom (Start-At (0,SCM+FSA)) by SCMFSA_2:103;
f <> intloc 0 by SCMFSA_2:58;
then not f in {(intloc 0)} by TARSKI:def 1;
then not f in dom ((intloc 0) .--> 1) ;
then A7: not f in dom (Initialize ((intloc 0) .--> 1)) by A5, A6, XBOOLE_0:def 3;
for m being Nat st m < n holds
IC (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),m)) in dom I

proof

IC (s +* (Initialize ((intloc 0) .--> 1))) = 0 by A3, MEMSTR_0:47;
then IC (s +* (Initialize ((intloc 0) .--> 1))) in dom I by AFINSQ_1:65;
hence for m being Nat st m < n holds
IC (Comput ((P +* I),(s +* (Initialize ((intloc 0) .--> 1))),m)) in dom I by A2, AMISTD_1:21; :: thesis: verum

end;

hence (IExec (I,P,s)) . f = (s +* (Initialize ((intloc 0) .--> 1))) . f by A1, A4, FUNCT_4:25, SF_MASTR:63
.= s . f by A7, FUNCT_4:11 ;
:: thesis: verum