reserve x for set,
  m,n for Nat,
  a,b,c for Int_position,
  i for Instruction of SCMPDS,
  s,s1,s2 for State of SCMPDS,
  k1,k2 for Integer,
  loc,l1 for Nat,
  I,J for Program of SCMPDS,
  N for with_non-empty_elements set;
reserve P,P1,P2,Q for Instruction-Sequence of SCMPDS;

theorem Th13:
  for s being 0-started State of SCMPDS
  for I being parahalting Program of SCMPDS,k be Nat st
    I c= P & k <= LifeSpan(P +* stop I,s)
   holds Comput(P, s,k) = Comput(P +* stop I,s,k)
proof
  let s be 0-started State of SCMPDS;
  let I be parahalting Program of SCMPDS,k be Nat;
  set m=LifeSpan(P +* stop I,s);
  assume that
A1: I c= P and
A2: k <= m;
  set s2 = s, P2 = P +* stop I;
  defpred P[Nat] means
    $1 <= m implies Comput(P,s,$1) = Comput(P2,s2,$1);
A3: P = P +* I by A1,FUNCT_4:98;
A4: now
    let k be Nat;
    assume
A5: P[k];
    now
A6:   Comput(P2,s2,k+1) = Following(P2,Comput(P2,s2,k)) by EXTPRO_1:3;
A7:   Comput(P,s,k+1) = Following(P,Comput(P,s,k)) by EXTPRO_1:3;
A8:  k < k+1 by XREAL_1:29;
      assume
A9:  k+1 <= m;
      then
A10:    k < m by A8,XXREAL_0:2;
      then IC Comput(P2,s2,k) in dom I by Th12;
      then
A11:  IC Comput(P2,s2,k) in dom (stop I) by FUNCT_4:12;
A12:   IC Comput(P2,s2,k) in dom I by A10,Th12;
      CurInstr(P,Comput(P,s,k))
         = P.IC Comput(P2,s2,k) by A5,A9,A8,PBOOLE:143,XXREAL_0:2
        .= I.IC Comput(P2,s2,k) by A3,A10,Th12,FUNCT_4:13
        .= (stop I).IC Comput(P2,s2,k) by A12,AFINSQ_1:def 3
        .= (P +* stop I).IC Comput(P2,s2,k) by A11,FUNCT_4:13
        .= CurInstr(P2,Comput(P2,s2,k)) by PBOOLE:143;
      hence  Comput(P,s,k+1) =  Comput(P2,s2,k+1)
        by A5,A9,A8,A7,A6,XXREAL_0:2;
    end;
    hence P[k+1];
  end;
A13: P[0];
  for k be Nat holds P[k] from NAT_1:sch 2(A13,A4);
  hence thesis by A2;
end;
