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 Program of SCMPDS,J being
  halt-free parahalting shiftable 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,J),P,Initialize s).b = IExec(J,P,Initialize s).b
proof
  let s be State of SCMPDS,I be Program of SCMPDS,J be halt-free parahalting
  shiftable 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+card J+2),SCMPDS) by SCMPDS_4:18;
  J is_closed_on s,P & J is_halting_on s,P by Th11,Th12;
  then
  IExec(if=0(a,k1,I,J),P,Initialize s)
   = IExec(J,P,Initialize s) +* Start-At((card I + card J
  + 2),SCMPDS) by A1,Th32;
  hence thesis by A2,FUNCT_4:11;
end;
