
theorem Th7:
  for L be reflexive transitive RelStr for X be Subset of L holds
  downarrow downarrow X = downarrow X
proof
  let L be reflexive transitive RelStr;
  let X be Subset of L;
A1: downarrow downarrow X c= downarrow X
  proof
    let x be object;
    assume
A2: x in downarrow downarrow X;
    then reconsider x1 = x as Element of L;
    consider y be Element of L such that
A3: y >= x1 and
A4: y in downarrow X by A2,WAYBEL_0:def 15;
    consider z be Element of L such that
A5: z >= y and
A6: z in X by A4,WAYBEL_0:def 15;
    z >= x1 by A3,A5,YELLOW_0:def 2;
    hence thesis by A6,WAYBEL_0:def 15;
  end;
  downarrow X c= downarrow downarrow X by WAYBEL_0:16;
  hence thesis by A1;
end;
