
theorem Th2:
  for T being TopSpace, K being Subset-Family of T holds K is Basis
  of T iff K \ {{}} is Basis of T
proof
  let T be TopSpace, K be Subset-Family of T;
  reconsider K9 = K \ {{}} as Subset-Family of T;
A1: UniCl K9 c= UniCl K by CANTOR_1:9,XBOOLE_1:36;
A2: K \ {{}} c= K by XBOOLE_1:36;
  hereby
    assume
A3: K is Basis of T;
    then the topology of T c= UniCl K by CANTOR_1:def 2;
    then
A4: the topology of T c= UniCl K9 by Th1;
    K c= the topology of T by A3,TOPS_2:64;
    then K \ {{}} c= the topology of T by A2;
    hence K \ {{}} is Basis of T by A4,CANTOR_1:def 2,TOPS_2:64;
  end;
  assume
A5: K \ {{}} is Basis of T;
  then
A6: K9 c= the topology of T by TOPS_2:64;
A7: K c= the topology of T
  proof
    let x be object;
    assume
A8: x in K;
    per cases;
    suppose
      x = {};
      hence thesis by PRE_TOPC:1;
    end;
    suppose
      x <> {};
      then not x in {{}} by TARSKI:def 1;
      then x in K9 by A8,XBOOLE_0:def 5;
      hence thesis by A6;
    end;
  end;
  the topology of T c= UniCl K9 by A5,CANTOR_1:def 2;
  then the topology of T c= UniCl K by A1;
  hence thesis by A7,CANTOR_1:def 2,TOPS_2:64;
end;
