
theorem Th3: :: Under 4.5.
  for L1 be algebraic lower-bounded LATTICE holds CLweight L1 =
  card the carrier of CompactSublatt L1
proof
  let L1 be algebraic lower-bounded LATTICE;
  set CB = the set of all card B1 where B1 is with_bottom CLbasis of L1 ;
  the carrier of CompactSublatt L1 is with_bottom CLbasis of L1 by WAYBEL23:71;
  then
A1: card the carrier of CompactSublatt L1 in CB;
  then
A2: meet CB c= card the carrier of CompactSublatt L1 by SETFAM_1:3;
  now
    let X be set;
    assume X in CB;
    then consider B1 be with_bottom CLbasis of L1 such that
A3: X = card B1;
    the carrier of CompactSublatt L1 c= B1 by WAYBEL23:71;
    hence card the carrier of CompactSublatt L1 c= X by A3,CARD_1:11;
  end;
  then card the carrier of CompactSublatt L1 c= meet CB by A1,SETFAM_1:5;
  hence thesis by A2,XBOOLE_0:def 10;
end;
