
theorem Th2:
  for L be non empty reflexive RelStr for x be Element of L holds
  compactbelow x is Subset of CompactSublatt L
proof
  let L be non empty reflexive RelStr;
  let x be Element of L;
  now
    let v be object;
    assume v in compactbelow x;
    then v in {z where z is Element of L: x >= z & z is compact} by
WAYBEL_8:def 2;
    then ex v1 be Element of L st v1 = v & x >= v1 & v1 is compact;
    hence v in the carrier of CompactSublatt L by WAYBEL_8:def 1;
  end;
  hence thesis by TARSKI:def 3;
end;
