reserve x for set,
  m,n for Nat,
  a,b for Int_position,
  i,j,k for Instruction of SCMPDS,
  s,s1,s2 for State of SCMPDS,
  k1,k2 for Integer,
  loc,l for Nat,
  I,J,K for Program of SCMPDS;
reserve P,P1,P2,Q for Instruction-Sequence of SCMPDS;

theorem Th40:
  for a be Int_position,i be Integer,n,m be Nat, I be
  Program of SCMPDS holds m < card I+3 iff  m in dom for-down(a,i,n,I)
proof
  let a be Int_position,i be Integer,n,m be Nat, I be Program of
  SCMPDS;
  card for-down(a,i,n,I)=card I + 3 by Th39;
  hence thesis by AFINSQ_1:66;
end;
