
theorem
  for L be up-complete non empty Poset st L is finite holds the
  carrier of L = the carrier of CompactSublatt L
proof
  let L be up-complete non empty Poset;
  assume
A1: L is finite;
A2: the carrier of L c= the carrier of CompactSublatt L
  proof
    let z be object;
    assume z in the carrier of L;
    then reconsider z1 = z as Element of L;
    z1 is compact by A1,WAYBEL_3:17;
    hence thesis by WAYBEL_8:def 1;
  end;
  the carrier of CompactSublatt L c= the carrier of L by YELLOW_0:def 13;
  hence thesis by A2;
end;
