reserve x for set,
  m,n for Nat,
  a,b for Int_position,
  i,j,k for Instruction of SCMPDS,
  s,s1,s2 for State of SCMPDS,
  k1,k2 for Integer,
  loc,l for Nat,
  I,J,K for Program of SCMPDS;
reserve P,P1,P2,Q for Instruction-Sequence of SCMPDS;

theorem Th19:
  for I,J being Program of SCMPDS,k be Nat st k <=
  LifeSpan(P +* stop I,Initialize s) & I c= J &
   I is_closed_on s,P & I is_halting_on s,P
   holds IC Comput(P +* J,Initialize s,k) in dom stop I
proof
  let I,J be Program of SCMPDS,k be Nat;
  set ss = Initialize s, PP = P +* stop I;
  set s1= Comput(P +* J, Initialize s,k),
s2= Comput(PP, ss,k);
  assume that
A1: k <= LifeSpan(PP,ss) and
A2: I c= J and
A3: I is_closed_on s,P and
A4: I is_halting_on s,P;
   s1 =  s2 by A1,A2,A3,A4,Th18;
  hence thesis by A3,SCMPDS_6:def 2;
end;
