reserve x for Int_position,
  n,p0 for Nat;
reserve P,Q,V for Instruction-Sequence of SCMPDS;

theorem Th1: :: see SCMPDS_6:87
  for s being 0-started State of SCMPDS,I being halt-free shiftable Program
  of SCMPDS, J being shiftable Program of SCMPDS,a,b be Int_position,k1 being
  Integer st s.DataLoc(s.a,k1) > 0 & I is_closed_on s,P & I is_halting_on s,P
   holds
  IExec(if>0(a,k1,I,J),P,s).b = IExec(I,P,s).b
proof
  let s be 0-started State of SCMPDS,
      I be halt-free shiftable Program of SCMPDS, J be
  shiftable Program of SCMPDS,a,b be Int_position,k1 be Integer;
  assume that
A1: s.DataLoc(s.a,k1)>0 and
A2: I is_closed_on s,P and
A3: I is_halting_on s,P;
A4: not b in dom Start-At((card I+card J+2),SCMPDS) by SCMPDS_4:18;
  IExec(if>0(a,k1,I,J),P,s) = IExec(I,P,s) +*
   Start-At((card I+card J+2),SCMPDS)
   by A1,A2,A3,SCMPDS_6:70;
  hence thesis by A4,FUNCT_4:11;
end;
