
theorem
  for T being non empty TopSpace, K being prebasis of T, x being Point of T
  for N being net of T
  st for A being Subset of T st A in K & x in A holds N is_eventually_in A
  for S being Subset of T st rng netmap(N,T) c= S holds x in Cl S
proof
  let T be non empty TopSpace, BB be prebasis of T, x be Point of T,
  N be net of T such that
A1: for A being Subset of T st A in BB & x in A holds N is_eventually_in A;
  let S be Subset of T such that
A2: rng netmap(N,T) c= S;
A3: [#]N is directed by WAYBEL_0:def 6;
  now
    let Z be finite Subset-Family of T;
    assume that
A4: Z c= BB and
A5: x in Intersect Z;
    defpred P[object,object] means
    for i,j being Element of N st $2 = i & i <= j
      ex pp being set st pp = $1 &  N.j in pp;
A6: now
      let a be object;
      assume
A7:   a in Z;
      then reconsider A = a as Subset of T;
      x in A by A5,A7,SETFAM_1:43;
      then N is_eventually_in A by A1,A4,A7;
      then consider i being Element of N such that
A8:   for j being Element of N st i <= j holds N.j in A;
      reconsider b = i as object;
      take b;
      thus b in the carrier of N & P[a, b] by A8;
    end;
    consider f being Function such that
A9: dom f = Z & rng f c= the carrier of N &
    for a being object st a in Z holds P[a, f.a]
    from FUNCT_1:sch 6(A6);
    set k = the Element of N;
    per cases by A9,RELAT_1:42;
    suppose Z = {};
      then
A10:  Intersect Z = the carrier of T by SETFAM_1:def 9;
      N.k in rng netmap(N,T) by FUNCT_2:4;
      hence Intersect Z meets S by A2,A10,XBOOLE_0:3;
    end;
    suppose rng f <> {};
      then reconsider Y = rng f as finite non empty Subset of N
      by A9,FINSET_1:8;
      consider i being Element of N such that
      i in the carrier of N and
A11:  i is_>=_than Y by A3,WAYBEL_0:1;
      now
        let y be set;
        assume
A12:    y in Z;
        then
A13:    f.y in Y by A9,FUNCT_1:def 3;
        then reconsider j = f.y as Element of N;
A14:     j <= i by A11,A13,LATTICE3:def 9;
        P[y,j] by A9,A12;
        then ex pp being set st pp = y &  N.i in pp by A14;
        hence N.i in y;
      end;
      then
A15:  N.i in Intersect Z by SETFAM_1:43;
      N.i in rng netmap(N,T) by FUNCT_2:4;
      hence Intersect Z meets S by A2,A15,XBOOLE_0:3;
    end;
  end;
  hence thesis by Th38;
end;
