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;
