
theorem Th6: :: PROPOSITION 4.6 (i)
  for T be T_0 non empty TopSpace for L1 be continuous
  lower-bounded sup-Semilattice st InclPoset the topology of T = L1 holds T is
  infinite implies weight T = CLweight L1
proof
  let T be T_0 non empty TopSpace;
  let L1 be continuous lower-bounded sup-Semilattice;
  assume that
A1: InclPoset the topology of T = L1 and
A2: T is infinite;
A3: the set of all  card B1 where B1 is Basis of T  c= the set of all  card B1
  where B1 is with_bottom CLbasis of L1
  proof
    let b be object;
    assume b in the set of all  card B1 where B1 is Basis of T ;
    then consider B2 be Basis of T such that
A4: b = card B2;
    B2 c= the topology of T by TOPS_2:64;
    then reconsider B3 = B2 as Subset of L1 by A1,YELLOW_1:1;
    B2 is infinite by A2,YELLOW15:30;
    then
A5: card B2 = card finsups B3 by YELLOW15:27;
    finsups B3 is with_bottom CLbasis of L1 by A1,Th5;
    hence thesis by A4,A5;
  end;
  the set of all  card B1 where B1 is with_bottom CLbasis of L1
  c= the set of all  card B1 where B1 is Basis of T
  proof
    let b be object;
    assume
    b in the set of all  card B1 where B1 is with_bottom CLbasis of L1 ;
    then consider B2 be with_bottom CLbasis of L1 such that
A6: b = card B2;
    B2 is Basis of T by A1,Th4;
    hence thesis by A6;
  end;
  then the set of all  card B1 where B1 is Basis of T  = the set of all
 card B1
  where B1 is with_bottom CLbasis of L1  by A3,
XBOOLE_0:def 10;
  hence thesis by WAYBEL23:def 5;
end;
