
theorem
  for L be non empty reflexive transitive RelStr for x be Element of L
  holds downarrow downarrow x = downarrow x
proof
  let L be non empty reflexive transitive RelStr;
  let x be Element of L;
A1: downarrow downarrow x c= downarrow x
  proof
    let v be object;
    assume
A2: v in downarrow downarrow x;
    then reconsider v1 = v as Element of L;
    consider y be Element of L such that
A3: y >= v1 and
A4: y in downarrow x by A2,WAYBEL_0:def 15;
    x >= y by A4,WAYBEL_0:17;
    then x >= v1 by A3,YELLOW_0:def 2;
    hence thesis by WAYBEL_0:17;
  end;
  downarrow x c= downarrow downarrow x by WAYBEL_0:16;
  hence thesis by A1;
end;
