
theorem
  for T be non empty TopSpace st T is finite for B be Basis of T for x
be Element of T holds meet { A where A is Element of the topology of T : x in A
  } in B
proof
  deffunc F(set) = $1;
  let T be non empty TopSpace;
  assume T is finite;
  then reconsider tT = the topology of T as finite non empty set;
  let B be Basis of T;
  let x be Element of T;
  defpred P[set] means x in $1;
  { A where A is Element of the topology of T : x in A } c= bool the
  carrier of T
  proof
    let z be object;
    assume z in { A where A is Element of the topology of T : x in A };
    then ex A be Element of the topology of T st z = A & x in A;
    hence thesis;
  end;
  then reconsider
  sfA = { A where A is Element of the topology of T : x in A } as
  Subset-Family of T;
  reconsider sfA as Subset-Family of T;
A1: now
    let Y be set;
    assume Y in sfA;
    then ex A be Element of the topology of T st Y = A & x in A;
    hence x in Y;
  end;
  the carrier of T is Element of the topology of T by PRE_TOPC:def 1;
  then the carrier of T in sfA;
  then
A2: x in meet sfA by A1,SETFAM_1:def 1;
A3: now
    let P be Subset of T;
    assume P in sfA;
    then ex A be Element of the topology of T st P = A & x in A;
    hence P is open by PRE_TOPC:def 2;
  end;
  { F(A) where A is Element of tT : P[A] } is finite from PRE_CIRC:sch 1;
  then meet sfA is open by A3,TOPS_2:20,def 1;
  then consider a be Subset of T such that
A4: a in B and
A5: x in a and
A6: a c= meet sfA by A2,YELLOW_9:31;
  meet sfA c= a
  proof
    let z be object;
    B c= the topology of T by TOPS_2:64;
    then a in sfA by A4,A5;
    hence thesis by SETFAM_1:def 1;
  end;
  hence thesis by A4,A6,XBOOLE_0:def 10;
end;
