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;
