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

theorem Th19: ::SCMPDS_5:32
  for I being halt-free Program of SCMPDS,s being State of
  SCMPDS, k being Nat st I is_closed_on s,P &
  I is_halting_on s,P & k <
  LifeSpan(P +* stop I,Initialize s)
   holds
  CurInstr(P +* stop I,Comput(P +* stop I,Initialize s,k)) <> halt SCMPDS
proof
  let I be halt-free Program of SCMPDS,s be State of SCMPDS, k be Nat;
  set ss=Initialize s, PP = P +* stop I,
      s2= Comput(PP, ss,k), P2 = PP;
  assume
  I is_closed_on s,P & I is_halting_on s,P & k < LifeSpan(PP,ss);
  then
A1: IC s2 in dom I by Th17;
A2:  P2/.IC s2 = P2.IC s2 by PBOOLE:143;
A3: stop I c= PP by FUNCT_4:25;
   I c= stop I by AFINSQ_1:74;
  then I c= PP by A3,XBOOLE_1:1;
  then CurInstr(P2,s2)=I.IC s2 by A1,A2,GRFUNC_1:2;
  hence thesis by A1,COMPOS_1:def 27;
end;
