
theorem
  for T being non empty TopStruct, P being Subset-Family of T st P c=
  the topology of T & for x being Point of T ex B being Basis of x st B c= P
  holds P is Basis of T
proof
  let T be non empty TopStruct;
  let P be Subset-Family of T such that
A1: P c= the topology of T and
A2: for x being Point of T ex B being Basis of x st B c= P;
  P is quasi_basis
  proof
  let e be object;
  assume
A3: e in the topology of T;
  then reconsider S = e as Subset of T;
  set X = { V where V is Subset of T: V in P & V c= S };
A4: X c= P
  proof
    let e be object;
    assume e in X;
    then ex V being Subset of T st e = V & V in P & V c= S;
    hence thesis;
  end;
  then reconsider X as Subset-Family of T by XBOOLE_1:1;
  for u being object holds u in S iff ex Z being set st u in Z & Z in X
  proof
    let u be object;
    hereby
      assume
A5:   u in S;
      then reconsider p = u as Point of T;
      consider B being Basis of p such that
A6:   B c= P by A2;
      S is open by A3;
      then consider W being Subset of T such that
A7:   W in B and
A8:   W c= S by A5,Def1;
      reconsider Z = W as set;
      take Z;
      thus u in Z by A7,Th12;
      thus Z in X by A6,A7,A8;
    end;
    given Z being set such that
A9: u in Z and
A10: Z in X;
    ex V being Subset of T st V = Z & V in P & V c= S by A10;
    hence thesis by A9;
  end;
  then S = union X by TARSKI:def 4;
  hence thesis by A4,CANTOR_1:def 1;
  end;
  hence thesis by A1,TOPS_2:64;
end;
