let s be State of ; :: thesis: for I being parahalting Program of st Initialized I c= s holds
for k being Element of NAT st k <= LifeSpan s holds
CurInstr (Computation (s +* (Directed I)),k) <> halt SCM+FSA
set A = NAT ;
let I be parahalting Program of ; :: thesis: ( Initialized I c= s implies for k being Element of NAT st k <= LifeSpan s holds
CurInstr (Computation (s +* (Directed I)),k) <> halt SCM+FSA )
set s2 = s +* (Directed I);
set m = LifeSpan s;
assume A1: Initialized I c= s ; :: thesis: for k being Element of NAT st k <= LifeSpan s holds
CurInstr (Computation (s +* (Directed I)),k) <> halt SCM+FSA
then A2: I +* (Start-At (insloc 0 )) c= s by Th8;
A3: ProgramPart s halts_on s by A1, AMI_1:def 26;

A4: now

set s1 = s +* (I ';' I);
let k be Element of NAT ; :: thesis: ( k <= LifeSpan s implies Computation s,k, Computation (s +* (Directed I)),k equal_outside NAT )
defpred S₁[ Element of NAT ] means ( $1 <= k implies Computation (s +* (I ';' I)),$1, Computation (s +* (Directed I)),$1 equal_outside NAT );
assume A5: k <= LifeSpan s ; :: thesis: Computation s,k, Computation (s +* (Directed I)),k equal_outside NAT
A6: for n being Element of NAT st S₁[n] holds
S₁[n + 1]

proof

A7: Directed I c= I ';' I by SCMFSA6A:55;
let n be Element of NAT ; :: thesis: ( S₁[n] implies S₁[n + 1] )
A8: dom I c= dom (I ';' I) by SCMFSA6A:56;
assume A9: ( n <= k implies Computation (s +* (I ';' I)),n, Computation (s +* (Directed I)),n equal_outside NAT ) ; :: thesis: S₁[n + 1]
A10: Computation (s +* (Directed I)),(n + 1) = Following (Computation (s +* (Directed I)),n) by AMI_1:14
.= Exec (CurInstr (Computation (s +* (Directed I)),n)),(Computation (s +* (Directed I)),n) ;
A11: Computation (s +* (I ';' I)),(n + 1) = Following (Computation (s +* (I ';' I)),n) by AMI_1:14
.= Exec (CurInstr (Computation (s +* (I ';' I)),n)),(Computation (s +* (I ';' I)),n) ;
A12: n <= n + 1 by NAT_1:12;
assume A13: n + 1 <= k ; :: thesis: Computation (s +* (I ';' I)),(n + 1), Computation (s +* (Directed I)),(n + 1) equal_outside NAT
then A14: IC (Computation (s +* (I ';' I)),n) = IC (Computation (s +* (Directed I)),n) by A9, A12, AMI_1:121, XXREAL_0:2;
n <= k by A13, A12, XXREAL_0:2;
then n <= LifeSpan s by A5, XXREAL_0:2;
then IC (Computation s,n) = IC (Computation (s +* (I ';' I)),n) by A3, A2, Th36, AMI_1:121;
then A15: IC (Computation (s +* (I ';' I)),n) in dom I by A2, Def2;
then A16: IC (Computation (s +* (Directed I)),n) in dom (Directed I) by A14, FUNCT_4:105;
A17: CurInstr (Computation (s +* (Directed I)),n) = (s +* (Directed I)) . (IC (Computation (s +* (Directed I)),n)) by AMI_1:54
.= (Directed I) . (IC (Computation (s +* (Directed I)),n)) by A16, FUNCT_4:14 ;
CurInstr (Computation (s +* (I ';' I)),n) = (s +* (I ';' I)) . (IC (Computation (s +* (I ';' I)),n)) by AMI_1:54
.= (I ';' I) . (IC (Computation (s +* (I ';' I)),n)) by A8, A15, FUNCT_4:14
.= (Directed I) . (IC (Computation (s +* (I ';' I)),n)) by A7, A14, A16, GRFUNC_1:8 ;
hence Computation (s +* (I ';' I)),(n + 1), Computation (s +* (Directed I)),(n + 1) equal_outside NAT by A9, A13, A12, A14, A17, A11, A10, SCMFSA6A:32, XXREAL_0:2; :: thesis: verum

end;

( Computation (s +* (I ';' I)),0 = s +* (I ';' I) & Computation (s +* (Directed I)),0 = s +* (Directed I) ) by AMI_1:13;
then Computation (s +* (Directed I)),0 , Computation (s +* (I ';' I)),0 equal_outside NAT by FUNCT_7:107, SCMFSA6A:42;
then A18: S₁[ 0 ] by FUNCT_7:28;
for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A18, A6);
then A19: Computation (s +* (I ';' I)),k, Computation (s +* (Directed I)),k equal_outside NAT ;
Computation s,k, Computation (s +* (I ';' I)),k equal_outside NAT by A3, A2, A5, Th36;
hence Computation s,k, Computation (s +* (Directed I)),k equal_outside NAT by A19, FUNCT_7:29; :: thesis: verum

end;

hereby :: thesis: verum

let k be Element of NAT ; :: thesis: ( k <= LifeSpan s implies not CurInstr (Computation (s +* (Directed I)),k) = halt SCM+FSA )
set lk = IC (Computation s,k);
A20: ( IC (Computation s,k) in dom I & dom I = dom (Directed I) ) by A2, Def2, FUNCT_4:105;
then A21: (Directed I) . (IC (Computation s,k)) in rng (Directed I) by FUNCT_1:def 5;
assume k <= LifeSpan s ; :: thesis: not CurInstr (Computation (s +* (Directed I)),k) = halt SCM+FSA
then IC (Computation s,k) = IC (Computation (s +* (Directed I)),k) by A4, AMI_1:121;
then A22: CurInstr (Computation (s +* (Directed I)),k) = (s +* (Directed I)) . (IC (Computation s,k)) by AMI_1:54
.= (Directed I) . (IC (Computation s,k)) by A20, FUNCT_4:14 ;
assume CurInstr (Computation (s +* (Directed I)),k) = halt SCM+FSA ; :: thesis: contradiction
hence contradiction by A22, A21, SCMFSA6A:18; :: thesis: verum

end;