
theorem :: REMARK 4.3
  for L be lower-bounded with_suprema non empty Poset holds
CompactSublatt L is join-inheriting & Bottom L in the carrier of CompactSublatt
  L
proof
  let L be lower-bounded with_suprema non empty Poset;
  now
    let x,y be Element of L;
    assume that
A1: x in the carrier of CompactSublatt L and
A2: y in the carrier of CompactSublatt L and
A3: ex_sup_of {x,y},L;
    y is compact by A2,Def1;
    then
A4: y << y by WAYBEL_3:def 2;
    x is compact by A1,Def1;
    then
A5: x << x by WAYBEL_3:def 2;
    y <= x "\/" y by A3,YELLOW_0:18;
    then
A6: y << x "\/" y by A4,WAYBEL_3:2;
    x <= x "\/" y by A3,YELLOW_0:18;
    then x << x "\/" y by A5,WAYBEL_3:2;
    then x "\/" y << x "\/" y by A6,WAYBEL_3:3;
    then x "\/" y is compact by WAYBEL_3:def 2;
    then sup {x,y} is compact by YELLOW_0:41;
    hence sup {x,y} in the carrier of CompactSublatt L by Def1;
  end;
  hence CompactSublatt L is join-inheriting by YELLOW_0:def 17;
  Bottom L is compact by WAYBEL_3:15;
  hence thesis by Def1;
end;
