
theorem
  for T being non empty TopSpace, P being Basis of T, p being Point of T
  holds {A where A is Subset of T: A in P & p in A} is Basis of p
proof
  let T be non empty TopSpace, P be Basis of T, p be Point of T;
  set Z = {A where A is Subset of T: A in P & p in A};
  Z c= bool the carrier of T
  proof
    let z be object;
    assume z in Z;
    then ex A being Subset of T st A = z & A in P & p in A;
    hence thesis;
  end;
  then reconsider Z as Subset-Family of T;
  reconsider Z as Subset-Family of T;
  Z is Basis of p
  proof
A1: Z is open
    proof
      let z be Subset of T;
      assume z in Z;
      then consider A being Subset of T such that
A2:   A = z and
A3:   A in P and
      p in A;
      A is open by A3,YELLOW_8:10;
      hence thesis by A2;
    end;
    Z is p-quasi_basis
    proof
    now
      let z be set;
      assume z in Z;
      then ex A being Subset of T st A = z & A in P & p in A;
      hence p in z;
    end;
    hence p in Intersect Z by SETFAM_1:43;
    let S be Subset of T;
    assume S is open & p in S;
    then consider V being Subset of T such that
A4: V in P & p in V & V c= S by YELLOW_9:31;
    take V;
    thus thesis by A4;
    end;
    hence thesis by A1;
  end;
  hence thesis;
end;
