
theorem Th1:
  for L be non empty Poset for x be Element of L holds compactbelow
  x = waybelow x /\ the carrier of CompactSublatt L
proof
  let L be non empty Poset;
  let x be Element of L;
A1: compactbelow x c= waybelow x /\ the carrier of CompactSublatt L
  proof
    let y be object;
    assume
A2: y in compactbelow x;
    then reconsider y1 = y as Element of L;
A3: y in downarrow x /\ the carrier of CompactSublatt L by A2,WAYBEL_8:5;
    then y in downarrow x by XBOOLE_0:def 4;
    then
A4: y1 <= x by WAYBEL_0:17;
A5: y in the carrier of CompactSublatt L by A3,XBOOLE_0:def 4;
    then y1 is compact by WAYBEL_8:def 1;
    then y1 << y1 by WAYBEL_3:def 2;
    then y1 << x by A4,WAYBEL_3:2;
    then y1 in waybelow x by WAYBEL_3:7;
    hence thesis by A5,XBOOLE_0:def 4;
  end;
  waybelow x /\ the carrier of CompactSublatt L c= compactbelow x
  proof
    let y be object;
    assume
A6: y in waybelow x /\ the carrier of CompactSublatt L;
    then reconsider y1 = y as Element of L;
    y in waybelow x by A6,XBOOLE_0:def 4;
    then y1 << x by WAYBEL_3:7;
    then y1 <= x by WAYBEL_3:1;
    then
A7: y in downarrow x by WAYBEL_0:17;
    y in the carrier of CompactSublatt L by A6,XBOOLE_0:def 4;
    then y in downarrow x /\ the carrier of CompactSublatt L by A7,
XBOOLE_0:def 4;
    hence thesis by WAYBEL_8:5;
  end;
  hence thesis by A1;
end;
