reserve A for set, x,y,z for object,
  k for Element of NAT;
reserve n for Nat,
  x for object;
reserve V, C for set;

theorem
  for X being set, A being empty Subset of X, R being Order of X st
  R linearly_orders A holds SgmX(R,A) = {}
proof
  let X be set, A be empty Subset of X, R be Order of X;
  assume R linearly_orders A;
  then rng SgmX(R,A) = {} by Def2;
  hence thesis;
end;
