
theorem
  for L be lower-bounded sup-Semilattice for B be Subset of L st B is
  infinite holds card B = card finsups B
proof
  let L be lower-bounded sup-Semilattice;
  let B be Subset of L;
  defpred P[Function, set] means $2 = "\/"(rng $1,L);
  assume
A1: B is infinite;
  then reconsider B1 = B as non empty Subset of L;
A2: for p be Element of B1* ex u be Element of finsups B1 st P[p, u]
  proof
    let p be Element of B1*;
A3: rng p c= the carrier of L by XBOOLE_1:1;
    now
      per cases;
      suppose
        rng p is empty;
        hence ex_sup_of rng p,L by YELLOW_0:42;
      end;
      suppose
        rng p is non empty;
        hence ex_sup_of rng p,L by A3,YELLOW_0:54;
      end;
    end;
    then "\/"(rng p,L) in { "\/"(Y,L) where Y is finite Subset of B1 :
    ex_sup_of Y,L};
    then reconsider u = "\/"(rng p,L) as Element of finsups B1 by
WAYBEL_0:def 27;
    take u;
    thus thesis;
  end;
  consider f be Function of B1*,finsups B1 such that
A4: for p be Element of B1* holds P[p, f.p] from FUNCT_2:sch 3(A2);
  B c= finsups B
  proof
    let z be object;
    assume
A5: z in B;
    then reconsider z1 = z as Element of L;
A6: {z1} c= B
    by A5,TARSKI:def 1;
    ex_sup_of {z1},L & z = sup {z1} by YELLOW_0:38,39;
    then z1 in { "\/"(Y,L) where Y is finite Subset of B : ex_sup_of Y,L } by
A6;
    hence thesis by WAYBEL_0:def 27;
  end;
  then
A7: card B c= card finsups B by CARD_1:11;
A8: dom f = B1* by FUNCT_2:def 1;
  finsups B c= rng f
  proof
    let z be object;
    assume z in finsups B;
    then z in { "\/"(Y,L) where Y is finite Subset of B : ex_sup_of Y,L } by
WAYBEL_0:def 27;
    then consider Y be finite Subset of B such that
A9: z = "\/"(Y,L) and
    ex_sup_of Y,L;
    consider p be FinSequence such that
A10: rng p = Y by FINSEQ_1:52;
    reconsider p as FinSequence of B1 by A10,FINSEQ_1:def 4;
    reconsider p1 = p as Element of B1* by FINSEQ_1:def 11;
    f.p1 = "\/"(rng p1,L) by A4;
    hence thesis by A8,A9,A10,FUNCT_1:def 3;
  end;
  then card finsups B1 c= card (B1*) by A8,CARD_1:12;
  then card finsups B1 c= card B1 by A1,CARD_4:24;
  hence thesis by A7;
end;
