
theorem Th13:
  for L being up-complete non empty Chain
  for x,y being Element of L st x < y holds x << y
proof
  let L be up-complete Chain, x,y be Element of L such that
A1: x <= y and
A2: x <> y;
  let D be non empty directed Subset of L such that
A3: y <= sup D and
A4: for d being Element of L st d in D holds not x <= d;
A5: ex_sup_of D,L by WAYBEL_0:75;
  x is_>=_than D
  proof
    let z be Element of L;
    assume z in D;
    then not x <= z by A4;
    hence thesis by WAYBEL_0:def 29;
  end;
  then x >= sup D by A5,YELLOW_0:def 9;
  then x >= y by A3,ORDERS_2:3;
  hence contradiction by A1,A2,ORDERS_2:2;
end;
