
theorem
  for L be non empty reflexive transitive RelStr for x be Element of L
  holds uparrow uparrow x = uparrow x
proof
  let L be non empty reflexive transitive RelStr;
  let x be Element of L;
A1: uparrow uparrow x c= uparrow x
  proof
    let v be object;
    assume
A2: v in uparrow uparrow x;
    then reconsider v1 = v as Element of L;
    consider y be Element of L such that
A3: y <= v1 and
A4: y in uparrow x by A2,WAYBEL_0:def 16;
    x <= y by A4,WAYBEL_0:18;
    then x <= v1 by A3,YELLOW_0:def 2;
    hence thesis by WAYBEL_0:18;
  end;
  uparrow x c= uparrow uparrow x by WAYBEL_0:16;
  hence thesis by A1;
end;
