let s be State of SCM+FSA; :: thesis: for P being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being parahalting Program of SCM+FSA st I c= P & Initialize ((intloc 0) .--> 1) c= s holds
for k being Element of NAT st k <= LifeSpan (P,s) holds
CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),s,k))) <> halt SCM+FSA
let P be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis: for I being parahalting Program of SCM+FSA st I c= P & Initialize ((intloc 0) .--> 1) c= s holds
for k being Element of NAT st k <= LifeSpan (P,s) holds
CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),s,k))) <> halt SCM+FSA
set A = NAT ;
let I be parahalting Program of SCM+FSA; :: thesis: ( I c= P & Initialize ((intloc 0) .--> 1) c= s implies for k being Element of NAT st k <= LifeSpan (P,s) holds
CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),s,k))) <> halt SCM+FSA )
set s2 = s;
set m = LifeSpan (P,s);
assume that
A1: I c= P and
A2: Initialize ((intloc 0) .--> 1) c= s ; :: thesis: for k being Element of NAT st k <= LifeSpan (P,s) holds
CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),s,k))) <> halt SCM+FSA
A3: Start-At (0,SCM+FSA) c= s by A2, LmA;
A5: P halts_on s by Def3, A1, A2, LmA;
set s1 = s;

A6: now

let k be Element of NAT ; :: thesis: ( k <= LifeSpan (P,s) implies NPP (Comput (P,s,k)) = NPP (Comput ((P +* (Directed I)),s,k)) )
defpred S₁[ Nat] means ( $1 <= k implies NPP (Comput ((P +* (I ';' I)),s,$1)) = NPP (Comput ((P +* (Directed I)),s,$1)) );
assume A7: k <= LifeSpan (P,s) ; :: thesis: NPP (Comput (P,s,k)) = NPP (Comput ((P +* (Directed I)),s,k))
A8: for n being Element of NAT st S₁[n] holds
S₁[n + 1]

proof

A9: Directed I c= I ';' I by SCMFSA6A:55;
let n be Element of NAT ; :: thesis: ( S₁[n] implies S₁[n + 1] )
A10: dom I c= dom (I ';' I) by SCMFSA6A:56;
assume A11: ( n <= k implies NPP (Comput ((P +* (I ';' I)),s,n)) = NPP (Comput ((P +* (Directed I)),s,n)) ) ; :: thesis: S₁[n + 1]
A12: Comput ((P +* (Directed I)),s,(n + 1)) = Following ((P +* (Directed I)),(Comput ((P +* (Directed I)),s,n))) by EXTPRO_1:4
.= Exec ((CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),s,n)))),(Comput ((P +* (Directed I)),s,n))) ;
A13: Comput ((P +* (I ';' I)),s,(n + 1)) = Following ((P +* (I ';' I)),(Comput ((P +* (I ';' I)),s,n))) by EXTPRO_1:4
.= Exec ((CurInstr ((P +* (I ';' I)),(Comput ((P +* (I ';' I)),s,n)))),(Comput ((P +* (I ';' I)),s,n))) ;
A14: n <= n + 1 by NAT_1:12;
assume A15: n + 1 <= k ; :: thesis: NPP (Comput ((P +* (I ';' I)),s,(n + 1))) = NPP (Comput ((P +* (Directed I)),s,(n + 1)))
then A16: IC (Comput ((P +* (I ';' I)),s,n)) = IC (Comput ((P +* (Directed I)),s,n)) by A11, A14, COMPOS_1:230, XXREAL_0:2;
n <= k by A15, A14, XXREAL_0:2;
then n <= LifeSpan (P,s) by A7, XXREAL_0:2;
then NPP (Comput (P,s,n)) = NPP (Comput ((P +* (I ';' I)),s,n)) by A5, A2, LmA, A1, Th36;
then IC (Comput (P,s,n)) = IC (Comput ((P +* (I ';' I)),s,n)) by COMPOS_1:230;
then A17: IC (Comput ((P +* (I ';' I)),s,n)) in dom I by Def2, A1, A2, LmA;
then A18: IC (Comput ((P +* (Directed I)),s,n)) in dom (Directed I) by A16, FUNCT_4:105;
dom (P +* (Directed I)) = NAT by PARTFUN1:def 4;
then A19: CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),s,n))) = (P +* (Directed I)) . (IC (Comput ((P +* (Directed I)),s,n))) by PARTFUN1:def 8
.= (Directed I) . (IC (Comput ((P +* (Directed I)),s,n))) by A18, FUNCT_4:14 ;
dom (P +* (I ';' I)) = NAT by PARTFUN1:def 4;
then CurInstr ((P +* (I ';' I)),(Comput ((P +* (I ';' I)),s,n))) = (P +* (I ';' I)) . (IC (Comput ((P +* (I ';' I)),s,n))) by PARTFUN1:def 8
.= (I ';' I) . (IC (Comput ((P +* (I ';' I)),s,n))) by A10, A17, FUNCT_4:14
.= (Directed I) . (IC (Comput ((P +* (I ';' I)),s,n))) by A9, A16, A18, GRFUNC_1:8 ;
hence NPP (Comput ((P +* (I ';' I)),s,(n + 1))) = NPP (Comput ((P +* (Directed I)),s,(n + 1))) by A11, A15, A14, A16, A19, A13, A12, AMISTD_2:def 20, XXREAL_0:2; :: thesis: verum

end;

( Comput ((P +* (I ';' I)),s,0) = s & Comput ((P +* (Directed I)),s,0) = s ) by EXTPRO_1:3;
then A20: S₁[ 0 ] ;
for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A20, A8);
then A21: NPP (Comput ((P +* (I ';' I)),s,k)) = NPP (Comput ((P +* (Directed I)),s,k)) ;
NPP (Comput (P,s,k)) = NPP (Comput ((P +* (I ';' I)),s,k)) by A5, A2, LmA, A7, Th36, A1;
hence NPP (Comput (P,s,k)) = NPP (Comput ((P +* (Directed I)),s,k)) by A21; :: thesis: verum

end;

hereby :: thesis: verum

let k be Element of NAT ; :: thesis: ( k <= LifeSpan (P,s) implies not CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),s,k))) = halt SCM+FSA )
set lk = IC (Comput (P,s,k));
A22: dom I = dom (Directed I) by FUNCT_4:105;
IC (Comput (P,s,k)) in dom I by Def2, A1, A2, LmA;
then A23: (Directed I) . (IC (Comput (P,s,k))) in rng (Directed I) by A22, FUNCT_1:def 5;
A24: dom (P +* (Directed I)) = NAT by PARTFUN1:def 4;
assume k <= LifeSpan (P,s) ; :: thesis: not CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),s,k))) = halt SCM+FSA
then NPP (Comput (P,s,k)) = NPP (Comput ((P +* (Directed I)),s,k)) by A6;
then IC (Comput (P,s,k)) = IC (Comput ((P +* (Directed I)),s,k)) by COMPOS_1:230;
then A25: CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),s,k))) = (P +* (Directed I)) . (IC (Comput (P,s,k))) by A24, PARTFUN1:def 8
.= (Directed I) . (IC (Comput (P,s,k))) by A22, Def2, A1, A3, FUNCT_4:14 ;
assume CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),s,k))) = halt SCM+FSA ; :: thesis: contradiction
hence contradiction by A25, A23, SCMFSA6A:18; :: thesis: verum

end;