reserve m, n for Nat,
  x for set,
  i for Instruction of SCM+FSA,
  I for Program of SCM+FSA,
  a for Int-Location,
  f for FinSeq-Location,
  l, l1 for Nat,
  s,s1,s2 for State of SCM+FSA,
  P,P1,P2 for Instruction-Sequence of SCM+FSA;

theorem Th13:
  for I being parahalting really-closed Program of SCM+FSA st
   I c= P & Initialize((intloc 0).-->1) c= s
holds for k being Nat st k <= LifeSpan(P,s)
 holds CurInstr(P+*Directed I,
 Comput(P+*Directed I,s,k)) <> halt SCM+FSA
proof
  let I be parahalting really-closed Program of SCM+FSA;
  set m = LifeSpan(P,s);
  assume that
A1: I c= P and
A2: Initialize((intloc 0).-->1) c= s;
A3: Start-At(0,SCM+FSA) c= s by A2,MEMSTR_0:50;
    then s is 0-started by MEMSTR_0:29;
    then
A4: P halts_on s by A1,AMISTD_1:def 11;
    reconsider s1 = s as 0-started State of SCM+FSA by A3,MEMSTR_0:29;
      IC s1 = 0 by MEMSTR_0:def 11;
      then
A5:    IC s in dom I by AFINSQ_1:65;
A6: now
    let k be Nat;
    defpred X[Nat] means $1 <= k implies
     Comput(P+*(I ";" I),s1,$1) =  Comput(P+*Directed I,s,$1);
    assume
A7: k <= m;
A8: for n being Nat st X[n] holds X[n+1]
    proof
A9:   Directed I c= I ";" I by SCMFSA6A:16;
      let n be Nat;
A10:   dom I c= dom (I ";" I) by SCMFSA6A:17;
      assume
A11:   n <= k implies
       Comput(P+*(I ";" I),s1,n) =  Comput(P+*Directed I,s,n);
A12:  Comput(P+*Directed I,s,n+1) =
Following(P+*Directed I,Comput(P+*Directed I,s,n))
 by EXTPRO_1:3
        .= Exec(CurInstr(P+*Directed I,Comput(P+*Directed I,s,n)),
        Comput(P+*Directed I,s,n));
A13:  Comput(P+*(I ";" I),s1,n+1) =
Following(P+*(I ";" I),Comput(P+*(I ";" I),s1,n)) by EXTPRO_1:3
        .= Exec(CurInstr(P+*(I ";" I),Comput(P+*(I ";" I),s1,n)),
        Comput(P+*(I ";" I),s1,n));
A14:  n <= n + 1 by NAT_1:12;
      assume
A15:  n + 1 <= k;
      n <= k by A15,A14,XXREAL_0:2;
      then  Comput(P,s,n) =  Comput(P+*(I ";" I),s1,n)
       by A4,A1,Th10,A7,XXREAL_0:2;
      then
A16:  IC Comput(P+*(I ";" I),s1,n) in dom I by A1,AMISTD_1:21,A5;
      then
A17:  IC Comput(P+*Directed I,s,n) in dom Directed I
         by A15,A11,A14,FUNCT_4:99,XXREAL_0:2;
 dom(P+*Directed I) = NAT by PARTFUN1:def 2;
    then
A18:  CurInstr(P+*Directed I,Comput(P+*Directed I,s,n))
 = (P+*Directed I).IC Comput(P+*Directed I,s,n) by PARTFUN1:def 6
        .= (Directed I).IC Comput(P+*Directed I,s,n) by A17,FUNCT_4:13;
    dom(P+*(I ";" I)) = NAT by PARTFUN1:def 2;
    then
      CurInstr(P+*(I ";" I),Comput(P+*(I ";" I),s1,n))
       = (P+*(I ";" I)).IC Comput(P+*(I ";" I),s1,n) by PARTFUN1:def 6
        .= (I ";" I).IC Comput(P+*(I ";" I),s1,n) by A10,A16,FUNCT_4:13
        .= (Directed I).IC Comput(P+*(I ";" I),s1,n)
         by A9,A15,A11,A14,A17,GRFUNC_1:2,XXREAL_0:2;
      hence thesis by A11,A15,A14,A18,A13,A12,XXREAL_0:2;
    end;
A19: X[0];
    for n being Nat holds X[n] from NAT_1:sch 2(A19,A8);
    then Comput(P+*(I ";" I),s1,k) =  Comput(P+*Directed I,s,k);
    hence  Comput(P,s,k) =  Comput(P+*Directed I,s,k) by A4,A7,Th10,A1;
  end;
    let k be Nat;
    set lk = IC Comput(P,s,k);
A20:  dom I = dom Directed I by FUNCT_4:99;
A21: IC Comput(P,s1,k) in dom I by A1,AMISTD_1:21,A5;
    then
A22: (Directed I).lk in rng Directed I by A20,FUNCT_1:def 3;
A23: dom(P+*Directed I) = NAT by PARTFUN1:def 2;
    assume k <= LifeSpan(P,s);
    then lk = IC Comput(P+*Directed I,s,k) by A6;
    then
A24: CurInstr(P+*Directed I,Comput(P+*Directed I,s,k))
      = (P+*Directed I).lk by A23,PARTFUN1:def 6
      .= (Directed I).lk by A20,A21,FUNCT_4:13;
    thus CurInstr(P+*Directed I,Comput(P+*Directed I,s,k)) <> halt SCM+FSA
      by A24,A22,SCMFSA6A:1;
end;
