
theorem Th15:
  for L being non empty RelStr, X being Subset of L holds uparrow 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 uparrow X;
    then reconsider y = x as Element of L;
    ex z being Element of L st z <= y & z in X by A1,Def16;
    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 Def16;
end;
