let s be State of SCM+FSA ; :: thesis: for I being Program of SCM+FSA st I is_closed_on Initialize s & I is_halting_on Initialize s holds
IExec (I ';' (Stop SCM+FSA )),s = (IExec I,s) +* (Start-At (insloc (card I)))
let I be Program of SCM+FSA ; :: thesis: ( I is_closed_on Initialize s & I is_halting_on Initialize s implies IExec (I ';' (Stop SCM+FSA )),s = (IExec I,s) +* (Start-At (insloc (card I))) )
assume A1: I is_closed_on Initialize s ; :: thesis: ( not I is_halting_on Initialize s or IExec (I ';' (Stop SCM+FSA )),s = (IExec I,s) +* (Start-At (insloc (card I))) )
assume A2: I is_halting_on Initialize s ; :: thesis: IExec (I ';' (Stop SCM+FSA )),s = (IExec I,s) +* (Start-At (insloc (card I)))
set s1 = s +* (Initialized I);
set s2 = s +* (Initialized (I ';' (Stop SCM+FSA )));
A3: s +* (Initialized I) = (Initialize s) +* (I +* (Start-At (insloc 0 ))) by Th13;
A4: dom (s | NAT ) = NAT by Th3;
A5: s +* (Initialized I) is halting by A2, A3, SCMFSA7B:def 8;
( s +* (Initialized (I ';' (Stop SCM+FSA ))) is halting & LifeSpan (s +* (Initialized (I ';' (Stop SCM+FSA )))) = (LifeSpan (s +* (Initialized I))) + 1 ) by A1, A2, Lm4;
then A6: Result (s +* (Initialized (I ';' (Stop SCM+FSA )))) = Computation (s +* (Initialized (I ';' (Stop SCM+FSA )))),((LifeSpan (s +* (Initialized I))) + 1) by AMI_1:122;
then DataPart (Result (s +* (Initialized (I ';' (Stop SCM+FSA ))))) = DataPart (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) by A1, A2, Lm4;
then A7: DataPart (Result (s +* (Initialized (I ';' (Stop SCM+FSA ))))) = DataPart (Result (s +* (Initialized I))) by A5, AMI_1:122
.= DataPart ((Result (s +* (Initialized I))) +* (Start-At (insloc (card I)))) by Th10 ;
IC (Result (s +* (Initialized (I ';' (Stop SCM+FSA ))))) = insloc (card I) by A1, A2, A6, Lm4
.= IC ((Result (s +* (Initialized I))) +* (Start-At (insloc (card I)))) by AMI_1:111 ;
then Result (s +* (Initialized (I ';' (Stop SCM+FSA )))),(Result (s +* (Initialized I))) +* (Start-At (insloc (card I))) equal_outside NAT by A7, Th6;
then A8: (Result (s +* (Initialized (I ';' (Stop SCM+FSA ))))) +* (s | NAT ) = ((Result (s +* (Initialized I))) +* (Start-At (insloc (card I)))) +* (s | NAT ) by A4, FUNCT_7:108;
A9: dom (s | NAT ) misses dom (Start-At (insloc (card I))) by Th12;
thus IExec (I ';' (Stop SCM+FSA )),s = (Result (s +* (Initialized (I ';' (Stop SCM+FSA ))))) +* (s | NAT ) by SCMFSA6B:def 1
.= (Result (s +* (Initialized I))) +* ((Start-At (insloc (card I))) +* (s | NAT )) by A8, FUNCT_4:15
.= (Result (s +* (Initialized I))) +* ((s | NAT ) +* (Start-At (insloc (card I)))) by A9, FUNCT_4:36
.= ((Result (s +* (Initialized I))) +* (s | NAT )) +* (Start-At (insloc (card I))) by FUNCT_4:15
.= (IExec I,s) +* (Start-At (insloc (card I))) by SCMFSA6B:def 1 ; :: thesis: verum