
theorem Th32:
  for T being TopSpace, B being Subset-Family of T st B c= the topology of T &
  for A being Subset of T st A is open for p being Point of T st p in A
  ex a being Subset of T st a in B & p in a & a c= A holds B is Basis of T
proof
  let T be TopSpace, B be Subset-Family of T such that
A1: B c= the topology of T and
A2: for A being Subset of T st A is open for p being Point of T st p in A
  ex a being Subset of T st a in B & p in a & a c= A;
A3:  B is open by A1,TOPS_2:64;
   B is quasi_basis
   proof
  let x be object;
  assume
A4: x in the topology of T;
  then reconsider A = x as Subset of T;
  set Y = {V where V is Subset of T: V in B & V c= A};
  Y c= bool the carrier of T
  proof
    let y be object;
    assume y in Y;
    then ex V being Subset of T st y = V & V in B & V c= A;
    hence thesis;
  end;
  then reconsider Y as Subset-Family of T;
A5: Y c= B
  proof
    let y be object;
    assume y in Y;
    then ex V being Subset of T st y = V & V in B & V c= A;
    hence thesis;
  end;
  A = union Y
  proof
    hereby
      let p be object;
      assume
A6:   p in A;
      then p in A;
      then reconsider q = p as Point of T;
      A is open by A4;
      then consider a being Subset of T such that
A7:   a in B and
A8:   q in a and
A9:   a c= A by A2,A6;
      a in Y by A7,A9;
      hence p in union Y by A8,TARSKI:def 4;
    end;
    let p be object;
    assume p in union Y;
    then consider a being set such that
A10: p in a and
A11: a in Y by TARSKI:def 4;
    ex V being Subset of T st a = V & V in B & V c= A by A11;
    hence thesis by A10;
  end;
  hence thesis by A5,CANTOR_1:def 1;
  end;
  hence thesis by A3;
end;
