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

A3: now

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

end;

now

assume A4: intloc 0 in NAT ; :: thesis: contradiction
then reconsider a = intloc 0 as Instruction-Location of SCM+FSA by AMI_1:def 4;
a in NAT by A4;
hence contradiction by SCMFSA_2:84; :: thesis: verum

end;

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 Initialize s ; :: thesis: ( (IExec I,s) . (intloc 0 ) = 1 & ( for k being Element of NAT holds (Computation ((Initialize s) +* (I +* (Start-At (insloc 0 )))),k) . (intloc 0 ) = 1 ) )
then A6: ProgramPart ((Initialize s) +* (I +* (Start-At (insloc 0 )))) halts_on (Initialize s) +* (I +* (Start-At (insloc 0 ))) by SCMFSA7B:def 8;
s +* (Initialized I) = (Initialize s) +* (I +* (Start-At (insloc 0 ))) by SCMFSA8A:13;
hence (IExec I,s) . (intloc 0 ) = ((Result ((Initialize s) +* (I +* (Start-At (insloc 0 ))))) +* (s | NAT )) . (intloc 0 ) by SCMFSA6B:def 1
.= (Result ((Initialize s) +* (I +* (Start-At (insloc 0 ))))) . (intloc 0 ) by A5, FUNCT_4:12
.= (Computation ((Initialize s) +* (I +* (Start-At (insloc 0 )))),(LifeSpan ((Initialize s) +* (I +* (Start-At (insloc 0 )))))) . (intloc 0 ) by A6, AMI_1:122
.= 1 by A3 ;
:: thesis: for k being Element of NAT holds (Computation ((Initialize s) +* (I +* (Start-At (insloc 0 )))),k) . (intloc 0 ) = 1
let k be Element of NAT ; :: thesis: (Computation ((Initialize s) +* (I +* (Start-At (insloc 0 )))),k) . (intloc 0 ) = 1
thus (Computation ((Initialize s) +* (I +* (Start-At (insloc 0 )))),k) . (intloc 0 ) = 1 by A3; :: thesis: verum