
theorem :: REMARK 4.2
  for L be complete LATTICE for x be Element of L holds uparrow x is
  Open Filter of L iff x is compact
proof
  let L be complete LATTICE;
  let x be Element of L;
  thus uparrow x is Open Filter of L implies x is compact
  proof
    x <= x;
    then
A1: x in uparrow x by WAYBEL_0:18;
    assume uparrow x is Open Filter of L;
    then consider y be Element of L such that
A2: y in uparrow x and
A3: y << x by A1,WAYBEL_6:def 1;
    x <= y by A2,WAYBEL_0:18;
    then x << x by A3,WAYBEL_3:2;
    hence thesis by WAYBEL_3:def 2;
  end;
  assume
A4: x is compact;
  now
    let u be Element of L;
    assume u in uparrow x;
    then
A5: x <= u by WAYBEL_0:18;
    take x2 = x;
    x <= x2;
    hence x2 in uparrow x by WAYBEL_0:18;
    x << x by A4,WAYBEL_3:def 2;
    hence x2 << u by A5,WAYBEL_3:2;
  end;
  hence thesis by WAYBEL_6:def 1;
end;
