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
   holds I is_closed_on s,P iff I is_closed_on Initialize s,P;
