
theorem
  for L1 be non empty Poset for A be Subset of L1 for a be Element of L1
  st a in uparrow A holds Way_Up(a,A) = {}
proof
  let L1 be non empty Poset;
  let A be Subset of L1;
  let a be Element of L1;
  assume
A1: a in uparrow A;
  wayabove a c= uparrow A
  proof
    let z be object;
    assume
A2: z in wayabove a;
    then reconsider z1 = z as Element of L1;
    a << z1 by A2,WAYBEL_3:8;
    then a <= z1 by WAYBEL_3:1;
    hence thesis by A1,WAYBEL_0:def 20;
  end;
  hence thesis by XBOOLE_1:37;
end;
