
theorem Th14:
  for L being non empty RelStr, X being Subset of L holds downarrow X =
  {x where x is Element of L: ex y being Element of L st x <= y & y in X}
proof
  let L be non empty RelStr, X be Subset of L;
  set Y={x where x is Element of L: ex y being Element of L st x <=
  y & y in X};
  hereby
    let x be object;
    assume
A1: x in downarrow X;
    then reconsider y = x as Element of L;
    ex z being Element of L st z >= y & z in X by A1,Def15;
    hence x in Y;
  end;
  let x be object;
  assume x in Y;
  then ex a being Element of L st a = x &
  ex y being Element of L st a <= y & y in X;
  hence thesis by Def15;
end;
