set a = intloc 0;
let s be State of SCM+FSA; :: thesis: for I being good Program of SCM+FSA st I is_halting_on Initialized s & I is_closed_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 A = NAT ;
let I be good Program of SCM+FSA; :: thesis: ( I is_halting_on Initialized s & I is_closed_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)));
defpred S₁[ Nat] means for n being Element of NAT st n <= $1 holds
(Comput ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))),n)) . (intloc 0) = (Initialized s) . (intloc 0);
assume I is_halting_on Initialized s ; :: thesis: ( not I is_closed_on Initialized s or ( (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 A1: ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))) halts_on (Initialized s) +* (I +* (Start-At (0,SCM+FSA))) by SCMFSA7B:def 8;
A2: S₁[ 0 ]

proof

let n be Element of NAT ; :: thesis: ( n <= 0 implies (Comput ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))),n)) . (intloc 0) = (Initialized s) . (intloc 0) )
A3: for i being Element of NAT st i < 0 holds
IC (Comput ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))),i)) in dom I ;
assume n <= 0 ; :: thesis: (Comput ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))),n)) . (intloc 0) = (Initialized s) . (intloc 0)
hence (Comput ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))),n)) . (intloc 0) = (Initialized s) . (intloc 0) by A3, Th95; :: thesis: verum

end;

assume A4: I is_closed_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 ) )
A5: for k being Element of NAT st S₁[k] holds
S₁[k + 1]

proof

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume S₁[k] ; :: thesis: S₁[k + 1]
let n be Element of NAT ; :: thesis: ( n <= k + 1 implies (Comput ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))),n)) . (intloc 0) = (Initialized s) . (intloc 0) )
assume A6: n <= k + 1 ; :: thesis: (Comput ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))),n)) . (intloc 0) = (Initialized s) . (intloc 0)
for i being Element of NAT st i < k + 1 holds
IC (Comput ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))),i)) in dom I by A4, SCMFSA7B:def 7;
hence (Comput ((ProgramPart ((Initialized s) +* (I +* (Start-At (0,SCM+FSA))))),((Initialized s) +* (I +* (Start-At (0,SCM+FSA)))),n)) . (intloc 0) = (Initialized s) . (intloc 0) by A6, Th95; :: thesis: verum

end;

A7: for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A2, A5);

A8: 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) . (intloc 0) by A7
.= 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 A10: not intloc 0 in dom (s | NAT) by RELAT_1:90;
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 A10, 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 A1, EXTPRO_1:23
.= 1 by A8 ;
:: 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
thus 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 by A8; :: thesis: verum