let I be Program of ; :: thesis: for s being State of st I is_closed_on Initialize s & I is_halting_on Initialize s holds
( IC (Computation (s +* (Initialized (I ';' (Stop SCM+FSA )))),((LifeSpan (s +* (Initialized I))) + 1)) = insloc (card I) & DataPart (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) = DataPart (Computation (s +* (Initialized (I ';' (Stop SCM+FSA )))),((LifeSpan (s +* (Initialized I))) + 1)) & ProgramPart (s +* (Initialized (I ';' (Stop SCM+FSA )))) halts_on s +* (Initialized (I ';' (Stop SCM+FSA ))) & LifeSpan (s +* (Initialized (I ';' (Stop SCM+FSA )))) = (LifeSpan (s +* (Initialized I))) + 1 )
let s be State of ; :: thesis: ( I is_closed_on Initialize s & I is_halting_on Initialize s implies ( IC (Computation (s +* (Initialized (I ';' (Stop SCM+FSA )))),((LifeSpan (s +* (Initialized I))) + 1)) = insloc (card I) & DataPart (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) = DataPart (Computation (s +* (Initialized (I ';' (Stop SCM+FSA )))),((LifeSpan (s +* (Initialized I))) + 1)) & ProgramPart (s +* (Initialized (I ';' (Stop SCM+FSA )))) halts_on s +* (Initialized (I ';' (Stop SCM+FSA ))) & LifeSpan (s +* (Initialized (I ';' (Stop SCM+FSA )))) = (LifeSpan (s +* (Initialized I))) + 1 ) )
assume A1: I is_closed_on Initialize s ; :: thesis: ( not I is_halting_on Initialize s or ( IC (Computation (s +* (Initialized (I ';' (Stop SCM+FSA )))),((LifeSpan (s +* (Initialized I))) + 1)) = insloc (card I) & DataPart (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) = DataPart (Computation (s +* (Initialized (I ';' (Stop SCM+FSA )))),((LifeSpan (s +* (Initialized I))) + 1)) & ProgramPart (s +* (Initialized (I ';' (Stop SCM+FSA )))) halts_on s +* (Initialized (I ';' (Stop SCM+FSA ))) & LifeSpan (s +* (Initialized (I ';' (Stop SCM+FSA )))) = (LifeSpan (s +* (Initialized I))) + 1 ) )
card (I ';' (Stop SCM+FSA )) = (card I) + 1 by Th17, SCMFSA6A:61;
then card I < card (I ';' (Stop SCM+FSA )) by NAT_1:13;
then A2: insloc (card I) in dom (I ';' (Stop SCM+FSA )) by SCMFSA6A:15;
A3: ProgramPart (Relocated (Stop SCM+FSA ),(card I)) c= Relocated (Stop SCM+FSA ),(card I) by RELAT_1:88;
A4: dom (ProgramPart (Stop SCM+FSA )) = dom (Stop SCM+FSA ) by AMI_1:105;
then insloc (0 + (card I)) in { (m + (card I)) where m is Element of NAT : m in dom (ProgramPart (Stop SCM+FSA )) } by Th15;
then A5: insloc (0 + (card I)) in dom (ProgramPart (Relocated (Stop SCM+FSA ),(card I))) by SCMFSA_5:3;
set s2 = s +* (Initialized (I ';' (Stop SCM+FSA )));
set s1 = s +* (Initialized I);
assume A6: I is_halting_on Initialize s ; :: thesis: ( IC (Computation (s +* (Initialized (I ';' (Stop SCM+FSA )))),((LifeSpan (s +* (Initialized I))) + 1)) = insloc (card I) & DataPart (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) = DataPart (Computation (s +* (Initialized (I ';' (Stop SCM+FSA )))),((LifeSpan (s +* (Initialized I))) + 1)) & ProgramPart (s +* (Initialized (I ';' (Stop SCM+FSA )))) halts_on s +* (Initialized (I ';' (Stop SCM+FSA ))) & LifeSpan (s +* (Initialized (I ';' (Stop SCM+FSA )))) = (LifeSpan (s +* (Initialized I))) + 1 )
then A7: IC (Computation (s +* (Initialized (Directed I))),((LifeSpan (s +* (Initialized I))) + 1)) = IC (Computation (s +* (Initialized (I ';' (Stop SCM+FSA )))),((LifeSpan (s +* (Initialized I))) + 1)) by A1, Th43;
I ';' (Stop SCM+FSA ) c= Initialized (I ';' (Stop SCM+FSA )) by SCMFSA6A:26;
then dom (I ';' (Stop SCM+FSA )) c= dom (Initialized (I ';' (Stop SCM+FSA ))) by GRFUNC_1:8;
then A8: (s +* (Initialized (I ';' (Stop SCM+FSA )))) . (insloc (card I)) = (Initialized (I ';' (Stop SCM+FSA ))) . (insloc (card I)) by A2, FUNCT_4:14
.= (I ';' (Stop SCM+FSA )) . (insloc (card I)) by A2, SCMFSA6A:50
.= (ProgramPart (Relocated (Stop SCM+FSA ),(card I))) . (insloc (card I)) by A5, FUNCT_4:14
.= (Relocated (Stop SCM+FSA ),(card I)) . (insloc (0 + (card I))) by A5, A3, GRFUNC_1:8
.= IncAddr (halt SCM+FSA ),(card I) by A4, Th16, SCMFSA_5:7, SCMNORM:2
.= halt SCM+FSA by SCMFSA_4:8 ;
DataPart (Computation (s +* (Initialized (Directed I))),((LifeSpan (s +* (Initialized I))) + 1)) = DataPart (Computation (s +* (Initialized (I ';' (Stop SCM+FSA )))),((LifeSpan (s +* (Initialized I))) + 1)) by A1, A6, Th43;
hence ( IC (Computation (s +* (Initialized (I ';' (Stop SCM+FSA )))),((LifeSpan (s +* (Initialized I))) + 1)) = insloc (card I) & DataPart (Computation (s +* (Initialized I)),(LifeSpan (s +* (Initialized I)))) = DataPart (Computation (s +* (Initialized (I ';' (Stop SCM+FSA )))),((LifeSpan (s +* (Initialized I))) + 1)) ) by A1, A6, A7, Th45; :: thesis: ( ProgramPart (s +* (Initialized (I ';' (Stop SCM+FSA )))) halts_on s +* (Initialized (I ';' (Stop SCM+FSA ))) & LifeSpan (s +* (Initialized (I ';' (Stop SCM+FSA )))) = (LifeSpan (s +* (Initialized I))) + 1 )
A9: CurInstr (Computation (s +* (Initialized (I ';' (Stop SCM+FSA )))),((LifeSpan (s +* (Initialized I))) + 1)) = (Computation (s +* (Initialized (I ';' (Stop SCM+FSA )))),((LifeSpan (s +* (Initialized I))) + 1)) . (insloc (card I)) by A1, A6, A7, Th45
.= halt SCM+FSA by A8, AMI_1:54 ;
hence A10: ProgramPart (s +* (Initialized (I ';' (Stop SCM+FSA )))) halts_on s +* (Initialized (I ';' (Stop SCM+FSA ))) by AMI_1:146; :: thesis: LifeSpan (s +* (Initialized (I ';' (Stop SCM+FSA )))) = (LifeSpan (s +* (Initialized I))) + 1

now

let k be Element of NAT ; :: thesis: ( k < (LifeSpan (s +* (Initialized I))) + 1 implies CurInstr (Computation (s +* (Initialized (I ';' (Stop SCM+FSA )))),k) <> halt SCM+FSA )
assume k < (LifeSpan (s +* (Initialized I))) + 1 ; :: thesis: CurInstr (Computation (s +* (Initialized (I ';' (Stop SCM+FSA )))),k) <> halt SCM+FSA
then A11: k <= LifeSpan (s +* (Initialized I)) by NAT_1:13;
then CurInstr (Computation (s +* (Initialized (Directed I))),k) <> halt SCM+FSA by A1, A6, Th44;
hence CurInstr (Computation (s +* (Initialized (I ';' (Stop SCM+FSA )))),k) <> halt SCM+FSA by A1, A6, A11, Th43; :: thesis: verum

end;

then for k being Element of NAT st CurInstr (Computation (s +* (Initialized (I ';' (Stop SCM+FSA )))),k) = halt SCM+FSA holds
(LifeSpan (s +* (Initialized I))) + 1 <= k ;
hence LifeSpan (s +* (Initialized (I ';' (Stop SCM+FSA )))) = (LifeSpan (s +* (Initialized I))) + 1 by A9, A10, AMI_1:def 46; :: thesis: verum