
theorem Th13:
  for L1 be finite LATTICE for B1 be with_bottom CLbasis of L1
  holds card B1 = CLweight L1 iff B1 = the carrier of CompactSublatt L1
proof
  let L1 be finite LATTICE;
  let B1 be with_bottom CLbasis of L1;
  thus card B1 = CLweight L1 implies B1 = the carrier of CompactSublatt L1
  proof
    assume card B1 = CLweight L1;
    then
A1: card the carrier of CompactSublatt L1 = card B1 by Th3;
    the carrier of CompactSublatt L1 c= B1 by WAYBEL23:71;
    hence thesis by A1,CARD_2:102;
  end;
  assume B1 = the carrier of CompactSublatt L1;
  hence thesis by Th3;
end;
