
theorem Th16:
  for L being reflexive RelStr, X being Subset of L holds
  X c= downarrow X & X c= uparrow X
proof
  let L be reflexive RelStr, X be Subset of L;
A1: the InternalRel of L is_reflexive_in the carrier of L by ORDERS_2:def 2;
  hereby
    let x be object;
    assume
A2: x in X;
    then reconsider y = x as Element of L;
    [y,y] in the InternalRel of L by A1,A2,RELAT_2:def 1;
    then y <= y by ORDERS_2:def 5;
    hence x in downarrow X by A2,Def15;
  end;
  let x be object;
  assume
A3: x in X;
  then reconsider y = x as Element of L;
  [y,y] in the InternalRel of L by A1,A3,RELAT_2:def 1;
  then y <= y by ORDERS_2:def 5;
  hence thesis by A3,Def16;
end;
