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, AMI_1:122
.= 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