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 Th15:
  for s being 0-started State of SCM+FSA
  for I being keeping_0 really-closed Program of SCM+FSA
   st not P+* I halts_on s
   for J being Program of SCM+FSA, k being Nat holds
    Comput(P+*I,s,k) =  Comput(P+*(I ";" J),s,k)
proof
  let s be 0-started State of SCM+FSA;
  let I be keeping_0 really-closed Program of SCM+FSA;
  assume
A1: not P+*I halts_on s;
  let J be Program of SCM+FSA;
  defpred X[Nat] means Comput(P+*I,s,$1) =  Comput(P+*(I ";" J),s,$1);
A2: for m st X[m] holds X[m+1]
  proof
    dom(I ";" J) = dom I \/ dom Reloc(J, card I) by SCMFSA6A:39;
    then
A3: dom I c= dom(I ";" J) by XBOOLE_1:7;
    let m;
A4: Comput(P+*I,s,m+1) = Following(P+*I,Comput(P+*I,s,m)) by EXTPRO_1:3
      .= Exec(CurInstr(P+*I,Comput(P+*I,s,m)),
      Comput(P+*I,s,m));
A5: Comput(P+*(I ";" J),s,m+1) =
Following(P+*(I ";" J),Comput(P+*(I ";" J),s,m))
 by EXTPRO_1:3
      .= Exec(CurInstr(P+*(I ";" J),
      Comput(P+*(I ";" J),s,m)),Comput(P+*(I ";" J),s,m));
   IC s = 0 by MEMSTR_0:def 11;
   then
A6: IC s in dom I by AFINSQ_1:65;
A7:   I c= P+*I by FUNCT_4:25;
    then
A8: IC Comput(P+*I,s,m) in dom I by AMISTD_1:21,A6;
    assume
A9:  Comput(P+*I,s,m) =  Comput(P+*(I ";" J),s,m);
   dom(P+*I) = NAT by PARTFUN1:def 2;
   then
A10:  (P+*I)/.IC Comput(P+*I,s,m)
 = (P+*I).IC Comput(P+*I,s,m) by PARTFUN1:def 6;
   dom(P+*(I ";" J)) = NAT by PARTFUN1:def 2;
   then
A11:  (P+*(I ";" J))/.IC Comput(P+*(I ";" J),s,m)
 = (P+*(I ";" J)).IC Comput(P+*(I ";" J),s,m) by PARTFUN1:def 6;
A12: I ";" J c= P +*(I ";" J) by FUNCT_4:25;
A13: CurInstr(P+*I,Comput(P+*I,s,m))
   = I.IC( Comput(P+*I,s,m)) by A8,A10,A7,GRFUNC_1:2;
    then I.IC( Comput(P+*I,s,m)) <> halt SCM+FSA by A1,EXTPRO_1:29;
    then CurInstr(P+*I,Comput(P+*I,s,m))
     = (I ";" J).IC( Comput(P+*I,s,m)) by A8,A13,SCMFSA6A:15
      .= CurInstr(P+*(I ";" J),Comput(P+*(I ";" J),s,m))
       by A9,A8,A3,A11,A12,GRFUNC_1:2;
    hence thesis by A9,A4,A5;
  end;
A14: X[0];
  thus for k being Nat holds X[k] from NAT_1:sch 2(A14, A2 );
end;
