
theorem Th19:
  for L1 be continuous lower-bounded sup-Semilattice st L1 is
  infinite for B1 be with_bottom CLbasis of L1 holds card { Way_Up(a,A) where a
  is Element of L1, A is finite Subset of L1 : a in B1 & A c= B1 } c= card B1
proof
  let L1 be continuous lower-bounded sup-Semilattice;
  assume
A1: L1 is infinite;
  let B1 be with_bottom CLbasis of L1;
  consider a1 be object such that
A2: a1 in B1 by XBOOLE_0:def 1;
  reconsider a1 as Element of L1 by A2;
  {}(L1) c= B1;
  then
  Way_Up(a1,{}(L1)) in { Way_Up(a,A) where a is Element of L1, A is finite
  Subset of L1 : a in B1 & A c= B1 } by A2;
  then reconsider
  WU = { Way_Up(a,A) where a is Element of L1, A is finite Subset
  of L1 : a in B1 & A c= B1 } as non empty set;
  defpred P[Element of B1*,set] means ex y be Element of L1, z be Subset of L1
  st y = $1/.1 & z = rng Del($1,1) & $2 = Way_Up(y,z);
A3: for x be Element of B1* ex u be Element of WU st P[x,u]
  proof
    let x be Element of B1*;
    reconsider y = x/.1 as Element of L1 by TARSKI:def 3;
    rng Del(x,1) c= rng x by FINSEQ_3:106;
    then
A4: rng Del(x,1) c= B1 by XBOOLE_1:1;
    then reconsider z = rng Del(x,1) as Subset of L1 by XBOOLE_1:1;
    Way_Up(y,z) in { Way_Up(a,A) where a is Element of L1, A is finite
    Subset of L1 : a in B1 & A c= B1 } by A4;
    then reconsider u = Way_Up(y,z) as Element of WU;
    take u,y,z;
    thus thesis;
  end;
  consider f be Function of B1*,WU such that
A5: for x be Element of B1* holds P[x,f.x] from FUNCT_2:sch 3(A3);
A6: dom f = B1* by FUNCT_2:def 1;
A7: WU c= rng f
  proof
    let z be object;
    assume z in WU;
    then consider a be Element of L1, A be finite Subset of L1 such that
A8: z = Way_Up(a,A) and
A9: a in B1 and
A10: A c= B1;
    reconsider a1 = a as Element of B1 by A9;
    consider p be FinSequence such that
A11: A = rng p by FINSEQ_1:52;
    reconsider p as FinSequence of B1 by A10,A11,FINSEQ_1:def 4;
    set q = <*a1*>^p;
A12: q in B1* by FINSEQ_1:def 11;
    then consider y be Element of L1, v be Subset of L1 such that
A13: y = q/.1 and
A14: v = rng Del(q,1) and
A15: f.q = Way_Up(y,v) by A5;
A16: a = y by A13,FINSEQ_5:15;
    A = v by A11,A14,WSIERP_1:40;
    hence thesis by A6,A8,A12,A15,A16,FUNCT_1:def 3;
  end;
  B1 is infinite by A1,Th17;
  then card B1 = card (B1*) by CARD_4:24;
  hence thesis by A6,A7,CARD_1:12;
end;
