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
  for s being State of SCMPDS,I being halt-free shiftable parahalting
Program of SCMPDS,a,b being Int_position,k1 being Integer st s.DataLoc(s.a,k1)
  <= 0 holds IExec(if<=0(a,k1,I),P,Initialize s).b = IExec(I,P,Initialize s).b
proof
  let s be State of SCMPDS,I be halt-free shiftable parahalting Program of
  SCMPDS,a,b be Int_position,k1 be Integer;
  assume
A1: s.DataLoc(s.a,k1) <= 0;
A2: not b in dom Start-At((card I+2),SCMPDS) by SCMPDS_4:18;
  I is_closed_on s,P & I is_halting_on s,P by Th11,Th12;
  then IExec(if<=0(a,k1,I),P,Initialize s) = IExec(I,P,Initialize s)
   +* Start-At((card I+2),SCMPDS)
   by A1,Th81;
  hence thesis by A2,FUNCT_4:11;
end;
