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

theorem :: SCMPDS_7:43
  for s being State of SCMPDS, I being parahalting Program of SCMPDS,J
  being shiftable Program of SCMPDS st
  J is_closed_on IExec(I,P,Initialize s),P & J
is_halting_on IExec(I,P,Initialize s),P holds I ';' J is_closed_on s,P
 & I ';' J is_halting_on s,P
proof
  let s be State of SCMPDS,I be parahalting Program of SCMPDS,J be shiftable
  Program of SCMPDS;
A1: I is_closed_on s,P & I is_halting_on s,P by SCMPDS_6:20,21;
  assume
  J is_closed_on IExec(I,P,Initialize s),P &
  J is_halting_on IExec(I,P,Initialize s),P;
  hence thesis by A1,SCMPDS_7:24;
end;
