
theorem Th9: :: PROPOSITION 4.6 (ii) (c)
  for T be TopStruct for L1 be continuous lower-bounded LATTICE st
  InclPoset the topology of T = L1 & T is finite holds CLweight L1 = card the
  carrier of L1
proof
  let T be TopStruct;
  let L1 be continuous lower-bounded LATTICE;
  assume
A1: InclPoset the topology of T = L1;
  [#]L1 is with_bottom CLbasis of L1 by YELLOW15:25;
  then
A2: card the carrier of L1 in the set of all
 card B1 where B1 is with_bottom CLbasis of
L1;
A3: CLweight L1 c= card the carrier of L1
  by A2,SETFAM_1:def 1;
  assume
A4: T is finite;
  now
    let Z be set;
    assume
    Z in the set of all  card B1 where B1 is with_bottom CLbasis of L1 ;
    then consider B1 be with_bottom CLbasis of L1 such that
A5: Z = card B1;
    Bottom L1 in B1 by WAYBEL23:def 8;
    then the carrier of CompactSublatt L1 c= B1 by WAYBEL23:48;
    then
A6: card the carrier of CompactSublatt L1 c= card B1 by CARD_1:11;
    L1 is finite by A1,A4,YELLOW_1:1;
    hence card the carrier of L1 c= Z by A5,A6,YELLOW15:26;
  end;
  then card the carrier of L1 c= meet the set of all
 card B1 where B1 is with_bottom
  CLbasis of L1  by A2,SETFAM_1:5;
  hence thesis by A3,XBOOLE_0:def 10;
end;
