
theorem Th2:
  for L1 be continuous sup-Semilattice ex B1 be with_bottom CLbasis
  of L1 st card B1 = CLweight L1
proof
  let L1 be continuous sup-Semilattice;
  defpred P[Ordinal] means $1 in the set of all
card B1 where B1 is with_bottom CLbasis of
L1;
  set X = the set of all  card B2 where B2 is with_bottom CLbasis of L1 ;
A1: ex A be Ordinal st P[A]
  proof
    take card [#]L1;
    [#]L1 is CLbasis of L1 by YELLOW15:25;
    hence thesis;
  end;
  consider A be Ordinal such that
A2: P[A] and
A3: for C be Ordinal st P[C] holds A c= C from ORDINAL1:sch 1(A1);
  consider B1 be with_bottom CLbasis of L1 such that
A4: A = card B1 by A2;
A5: now
    let x be object;
    thus (for y be set holds y in X implies x in y) implies x in A by A2;
    assume
A6: x in A;
    let y be set;
    assume
A7: y in X;
    then ex B2 be with_bottom CLbasis of L1 st y = card B2;
    then reconsider y1 = y as Cardinal;
    A c= y1 by A3,A7;
    hence x in y by A6;
  end;
  take B1;
  [#]L1 is CLbasis of L1 by YELLOW15:25;
  then card [#]L1 in X;
  hence thesis by A4,A5,SETFAM_1:def 1;
end;
