reserve m,n for Nat,
  I for Program of SCM+FSA,
  s,s1,s2 for State of SCM+FSA,
  a for Int-Location,
  f for FinSeq-Location,
  p,p1,p2 for Instruction-Sequence of SCM+FSA;

theorem Th11:
  for I being InitHalting really-closed Program of SCM+FSA st
    Initialize ((intloc 0) .--> 1) c= s & I c= p
  for k being Nat st k <= LifeSpan(p,s)
   holds CurInstr(p +* Directed I,
    Comput(p +* Directed I, s,k)) <> halt SCM+FSA
proof
  set A = NAT;
  let I be InitHalting really-closed Program of SCM+FSA;
  set s2 = s +* EP,
      p2 = p +* Directed I;
  set m = LifeSpan(p,s);
  assume
A1: iS c= s;
  assume
A2: I c= p;
   then
A3: p halts_on s by A1,Def1;
A4: now
    set s1 = s +* EP,
        p1 = p +* (I ";" I);
    let k be Nat;
    defpred X[Nat] means $1 <= k implies
     Comput(p1, s1,$1) =  Comput(p2, s2,$1);
    assume
A5: k <= m;
A6: for n being Nat st X[n] holds X[n+1]
    proof
A7:   Directed I c= I ";" I by SCMFSA6A:16;
      let n be Nat;
A8:   dom I c= dom (I ";" I) by SCMFSA6A:17;
      assume
A9:   n <= k implies
       Comput(p1, s1,n) =  Comput(p2,s2,n);
A10:   Comput(p2, s2,n + 1) = Following(p2,Comput(p2,s2,n))
 by EXTPRO_1:3
        .= Exec(CurInstr(p2,Comput(p2,s2,n)),Comput(p2, s2,n));
A11:  Comput(p1, s1,n + 1) = Following(p1,Comput(p1,s1,n)) by EXTPRO_1:3
        .= Exec(CurInstr(p1,Comput(p1,s1,n)),Comput(p1, s1,n));
A12:  n <= n + 1 by NAT_1:12;
      assume
A13:  n + 1 <= k;
     IC s1 = 0 by A1,MEMSTR_0:47;
     then
A14:   IC s1 in dom I by AFINSQ_1:65;
      n <= k by A13,A12,XXREAL_0:2;
      then Comput(p,s,n) = Comput(p1, s1,n) by A1,A3,Th8,A2,A5,XXREAL_0:2;
      then
A15:  IC Comput(p1, s1,n) in dom I by AMISTD_1:21,A2,A14;
      then
A16:  IC Comput(p2, s2,n) in dom Directed I
          by A13,A9,A12,FUNCT_4:99,XXREAL_0:2;
A17:  CurInstr(p2,Comput(p2,s2,n))
        = p2.IC Comput(p2, s2,n) by PBOOLE:143
        .= (Directed I).IC Comput(p2, s2,n) by A16,FUNCT_4:13;
      CurInstr(p1,Comput(p1,s1,n))
       = p1.IC Comput(p1, s1,n) by PBOOLE:143
        .= (I ";" I).IC Comput(p1, s1,n) by A8,A15,FUNCT_4:13
        .= (Directed I).IC Comput(p1, s1,n)
          by A7,A13,A16,A9,A12,GRFUNC_1:2,XXREAL_0:2;
      hence thesis by A9,A13,A12,A17,A11,A10,XXREAL_0:2;
    end;
A18: X[0];
    for n being Nat holds X[n] from NAT_1:sch 2(A18,A6);
    then Comput(p1,s1,k) =  Comput(p2,s2,k);
    hence  Comput(p,s,k) =  Comput(p2,s2,k) by A1,A3,A5,Th8,A2;
  end;
    let k be Nat;
    set lk = IC Comput(p,s,k);
     IC s = 0 by A1,MEMSTR_0:47;
     then IC s in dom I by AFINSQ_1:65;
     then
A19: IC Comput(p,s,k) in dom I &
     dom I = dom Directed I by A2,AMISTD_1:21,
FUNCT_4:99;
    then
A20: (Directed I).lk in rng Directed I by FUNCT_1:def 3;
    assume k <= LifeSpan(p,s);
    then lk = IC Comput(p2,s2,k) by A4;
    then
 A21: CurInstr(p2,Comput(p2,s2,k))
       = p2.lk by PBOOLE:143
      .= (Directed I).lk by A19,FUNCT_4:13;
    assume
    CurInstr(p +* Directed I,
     Comput(p +* Directed I, s,k)) = halt SCM+FSA;
    hence contradiction by A21,A20,SCMFSA6A:1;
end;
