
theorem :: WAYBEL31:1, AK, 21.02.2006
  for T be TopStruct ex b be Basis of T st card b = weight T
proof
  let T be TopStruct;
  defpred P[Ordinal] means $1 in the set of all  card b where b is Basis of T ;
  set X = the set of all  card b1 where b1 is Basis of T ;
A1: ex A be Ordinal st P[A]
  proof
    take card the topology of T;
    the topology of T is Basis of T by CANTOR_1:2;
    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 b be Basis of T such that
A4: A = card b 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 Basis of T st y = card B2;
    then reconsider y1 = y as Cardinal;
    A c= y1 by A3,A7;
    hence x in y by A6;
  end;
  take b;
  the topology of T is Basis of T by CANTOR_1:2;
  then card the topology of T in X;
  hence thesis by A4,A5,SETFAM_1:def 1;
end;
