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 Th10:
  for I being really-closed Program of SCM+FSA
   st p+*I halts_on s & Directed I c= p &
 Initialize ((intloc 0) .--> 1) c= s
 holds DataPart Comput(p, s,LifeSpan(p +* I,s))
  = DataPart Comput(p,s,LifeSpan(p +* I,s) + 1)
proof
  set A = NAT;
  let I be really-closed Program of SCM+FSA;
  assume that
A1: p+*I halts_on s and
A2: Directed I c= p and
A3: iS c= s;
  set sISA0 = s +* iS,
      pISA0 = p +* I;
  set s2 = sISA0 +* EP,
      p2 = pISA0 +* Directed I;
A4: iS c= sISA0 by FUNCT_4:25;
A5: I c= p+*I by FUNCT_4:25;
A6: sISA0 = s by A3,FUNCT_4:98;
  reconsider sISA0 as State of SCM+FSA;
  set m = LifeSpan(pISA0,sISA0);
  set l1 = IC Comput(pISA0, sISA0,m);
   IC sISA0 = 0 by MEMSTR_0:def 11;
   then IC sISA0 in dom I by AFINSQ_1:65;
   then
A7: l1 in dom I by AMISTD_1:21,A5;
  set s2 = sISA0 +* EP,
      p2 = pISA0 +* Directed I;
 now
    set s1 = sISA0 +* EP,
        p1 = pISA0 +* (I ";" I);
    let k be Nat;
    defpred X[Nat] means $1 <= k implies
      Comput(p1, s1,$1) =  Comput(p2, s2,$1);
    assume
A8: k <= m;
A9: for n being Nat st X[n] holds X[n+1]
    proof
A10:  Directed I c= I ";" I by SCMFSA6A:16;
      let n be Nat;
A11:  dom I c= dom (I ";" I) by SCMFSA6A:17;
      assume
A12:  n <= k implies
         Comput(p1, s1,n) =  Comput(p2,s2,n);
A13:  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));
A14:  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));
A15:  n <= n + 1 by NAT_1:12;
      assume
A16:  n + 1 <= k;
      IC s1 = 0 by MEMSTR_0:def 11;
      then
A17:    IC s1 in dom I by AFINSQ_1:65;
      n <= k by A16,A15,XXREAL_0:2;
      then  Comput(pISA0, sISA0,n) =  Comput(p1, s1,n)
       by A1,A4,Th8,A5,A6,A8,XXREAL_0:2;
      then
A18:  IC Comput(p1, s1,n) in dom I by AMISTD_1:21,A5,A17;
      then
A19:  IC Comput(p2, s2,n) in dom Directed I
        by A16,A12,A15,FUNCT_4:99,XXREAL_0:2;
A20:  CurInstr(p2,Comput(p2,s2,n))
         = p2.IC Comput(p2, s2,n) by PBOOLE:143
        .= (Directed I).IC Comput(p2, s2,n) by A19,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 A11,A18,FUNCT_4:13
        .= (Directed I).IC Comput(p1, s1,n)
         by A10,A16,A19,A12,A15,GRFUNC_1:2,XXREAL_0:2;
      hence thesis by A12,A16,A15,A20,A14,A13,XXREAL_0:2;
    end;
A21: X[0];
    for n being Nat holds X[n] from NAT_1:sch 2(A21,A9);
    then Comput(p1,s1,k) =  Comput(p2,s2,k);
    hence Comput(pISA0, sISA0,k) =  Comput(p2,s2,k) by A1,A4,A6,A8,Th8,A5;
  end;
  then
A22:  Comput(pISA0, sISA0,m) =  Comput(p2,s2,m);
A23: I.l1 = pISA0.l1 by A7,A5,GRFUNC_1:2
    .=CurInstr(pISA0,Comput(pISA0,sISA0,m)) by PBOOLE:143
    .= halt SCM+FSA by A1,A6,EXTPRO_1:def 15;
  IC Comput(p2,s2,m) in dom Directed I by A7,A22,FUNCT_4:99;
  then
A24: p2.l1 = (Directed I).l1 by A22,FUNCT_4:13
    .= goto  card I by A7,A23,FUNCT_4:106;
  Comput(p2, s2,m + 1)
     = Following(p2,Comput(p2,s2,m)) by EXTPRO_1:3
    .= Exec(goto  card I,Comput(p2,s2,m))
     by A22,A24,PBOOLE:143;
  then
A25: ( for a being Int-Location holds Comput(p2, s2,m + 1).a =
Comput(p2, s2,m). a)& for f being FinSeq-Location holds Comput(
p2, s2,m + 1).f =
  Comput(p2, s2,m).f by SCMFSA_2:69;
    dom Directed I = dom I by FUNCT_4:99;
    then
     p +* I +* Directed I = p +* Directed I by FUNCT_4:74
     .= p by A2,FUNCT_4:98;
  hence thesis by A6,A25,SCMFSA_M:2;
end;
