reserve a,b,c for set;

theorem Th1:
  for T being non empty TopSpace, B being Basis of T for x being
  Element of T holds {U where U is Subset of T: x in U & U in B} is Basis of x
proof
  let T be non empty TopSpace;
  let B be Basis of T;
  let x be Element of T;
  set Bx = {U where U is Subset of T: x in U & U in B};
A1: Bx c= B
  proof
    let a be object;
    assume a in Bx;
    then ex U being Subset of T st a = U & x in U & U in B;
    hence thesis;
  end;
  then reconsider Bx as Subset-Family of T by XBOOLE_1:1;
  Bx is Basis of x
  proof
    B c= the topology of T by TOPS_2:64;
    then Bx c= the topology of T by A1;
    then
A2: Bx is open by TOPS_2:64;
    Bx is x-quasi_basis
    proof
    now
      let a;
      assume a in Bx;
      then ex U being Subset of T st a = U & x in U & U in B;
      hence x in a;
    end;
    hence x in Intersect Bx by SETFAM_1:43;
    let S be Subset of T such that
A3: S is open and
A4: x in S;
    consider V being Subset of T such that
A5: V in B and
A6: x in V and
A7: V c= S by A3,A4,YELLOW_9:31;
    V in Bx by A5,A6;
    hence thesis by A7;
  end;
  hence thesis by A2;
  end;
  hence thesis;
end;
