set a = intloc 0;
let s be State of SCM+FSA; :: thesis: for I being keeping_0 Program of SCM+FSA st I is_halting_on Initialized s holds
( (IExec (I,s)) . (intloc 0) = 1 & ( for k being Element of NAT holds (Comput ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))),k)) . (intloc 0) = 1 ) )
set D = Int-Locations \/ FinSeq-Locations;
set A = NAT ;
let I be keeping_0 Program of SCM+FSA; :: thesis: ( I is_halting_on Initialized s implies ( (IExec (I,s)) . (intloc 0) = 1 & ( for k being Element of NAT holds (Comput ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))),k)) . (intloc 0) = 1 ) ) )
set s0 = Initialized s;
set s1 = (Initialized s) +* (I +* (Start-At (0,SCM+FSA)));
A1: I +* (Start-At (0,SCM+FSA)) c= (Initialized s) +* (I +* (Start-At (0,SCM+FSA))) by FUNCT_4:26;
A2: DataPart (Initialized s) = DataPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))) by SCMFSA8A:11;

A3: now

let k be Element of NAT ; :: thesis: (Comput ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))),k)) . (intloc 0) = 1
thus (Comput ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))),k)) . (intloc 0) = ((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))) . (intloc 0) by A1, SCMFSA6B:def 4
.= (Initialized s) . (intloc 0) by A2, SCMFSA6A:38
.= 1 by SCMFSA6C:3 ; :: thesis: verum

end;

not intloc 0 in NAT by SCMFSA_2:84;
then not intloc 0 in (dom s) /\ NAT by XBOOLE_0:def 4;
then A5: not intloc 0 in dom (s | NAT) by RELAT_1:90;
assume I is_halting_on Initialized s ; :: thesis: ( (IExec (I,s)) . (intloc 0) = 1 & ( for k being Element of NAT holds (Comput ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))),k)) . (intloc 0) = 1 ) )
then A6: ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))) halts_on (Initialized s) +* (I +* (Start-At (0,SCM+FSA))) by SCMFSA7B:def 8;
s +* (Initialized I) = (Initialized s) +* (I +* (Start-At (0,SCM+FSA))) by SCMFSA8A:13;
hence (IExec (I,s)) . (intloc 0) = ((Result ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))))) +* (s | NAT)) . (intloc 0) by SCMFSA6B:def 1
.= (Result ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))))) . (intloc 0) by A5, FUNCT_4:12
.= (Comput ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))))))) . (intloc 0) by A6, EXTPRO_1:23
.= 1 by A3 ;
:: thesis: for k being Element of NAT holds (Comput ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))),k)) . (intloc 0) = 1
let k be Element of NAT ; :: thesis: (Comput ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))),k)) . (intloc 0) = 1
thus (Comput ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))),k)) . (intloc 0) = 1 by A3; :: thesis: verum