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 Th8:
  for I being really-closed Program of SCM+FSA, J being Program of
  SCM+FSA st Initialize ((intloc 0) .--> 1) c= s & I c= p &
   p halts_on s
   for m st m <= LifeSpan(p,s) holds
   Comput(p,s,m) =  Comput(p+*(I ";" J),s,m)
proof
  let I be really-closed Program of SCM+FSA, J be Program of SCM+FSA;
  assume that
A1: iS c= s and
A2: I c= p and
A3: p halts_on s;
  defpred X[Nat] means $1 <= LifeSpan(p,s) implies
    Comput(p,s,$1) =  Comput(p+*(I ";" J),s,$1);
A4: for m st X[m] holds X[m+1]
  proof
    set px = p+*(I ";" J);
    let m;
A5: I ";" J c= px by FUNCT_4:25;
    assume
A6: m <= LifeSpan(p,s) implies
       Comput(p,s,m) =  Comput(p+*(I ";" J),s,m);
    dom(I ";" J) = dom I \/ dom Reloc(J, card I) by SCMFSA6A:39;
    then
A7: {} c= Comput(px,s,m) & dom I c= dom(I ";" J) by XBOOLE_1:2,7;
A8: Comput(p,s,m+1) = Following(p,Comput(p,s,m)) by EXTPRO_1:3
      .= Exec(CurInstr(p,Comput(p,s,m)),Comput(p,s,m));
A9: Comput(px,s,m+1) = Following(px,Comput(px,s,m)) by EXTPRO_1:3
      .= Exec(CurInstr(px,Comput(px,s,m)),
      Comput(px,s,m));
    IC s = 0 by A1,MEMSTR_0:47;
    then IC s in dom I by AFINSQ_1:65;
    then
A10: IC Comput(p,s,m) in dom I by AMISTD_1:21,A2;
A11:  p/.IC Comput(p,s,m) = p.IC Comput(p,s,m) by PBOOLE:143;
A12: CurInstr(p,Comput(p,s,m))
     = I.IC(Comput(p,s,m)) by A10,A11,A2,GRFUNC_1:2;
    assume
A13: m+1 <= LifeSpan(p,s);
A14:  px/.IC Comput(px,s,m)
 = px.IC Comput(px,s,m) by PBOOLE:143;
    m < LifeSpan(p,s) by A13,NAT_1:13;
    then I.IC(Comput(p,s,m)) <> halt SCM+FSA by A3,A12,EXTPRO_1:def 15;
    then CurInstr(p,Comput(p,s,m))
     = (I ";" J).IC(Comput(p,s,m)) by A10,A12,SCMFSA6A:15
    .= CurInstr(px,Comput(px,s,m)) by A13,A10,A7,A14,A5,A6,GRFUNC_1:2,NAT_1:13;
    hence thesis by A6,A13,A8,A9,NAT_1:13;
  end;
A15: X[0];
  thus for m holds X[m] from NAT_1:sch 2(A15,A4);
end;
