
theorem Th25:
  for T being non empty TopSpace, K being prebasis of T for N
being net of T, p being Point of T st for A being Subset of T st p in A & A in
  K holds N is_eventually_in A holds p in Lim N
proof
  let T be non empty TopSpace, K be prebasis of T;
  let N be net of T, x be Point of T such that
A1: for A being Subset of T st x in A & A in K holds N is_eventually_in A;
  now
    defpred P[object,object] means
    ex D1 being set st D1 = $1 &
   for i,j being Element of N st i = $2 & j >= i
    holds N.j in D1;
    let A be a_neighborhood of x;
A2: Int A in the topology of T by PRE_TOPC:def 2;
    FinMeetCl K is Basis of T by YELLOW_9:23;
    then the topology of T = UniCl FinMeetCl K by YELLOW_9:22;
    then consider Y being Subset-Family of T such that
A3: Y c= FinMeetCl K and
A4: Int A = union Y by A2,CANTOR_1:def 1;
    x in Int A by CONNSP_2:def 1;
    then consider y being set such that
A5: x in y and
A6: y in Y by A4,TARSKI:def 4;
    consider Z being Subset-Family of T such that
A7: Z c= K and
A8: Z is finite and
A9: y = Intersect Z by A3,A6,CANTOR_1:def 3;
A10: for a being object st a in Z
   ex b being object st b in the carrier of N & P[a,b]
    proof
      let a be object;
      assume
A11:  a in Z;
      then reconsider a as Subset of T;
      x in a by A5,A9,A11,SETFAM_1:43;
      then N is_eventually_in a by A1,A7,A11;
      then consider i being Element of N such that
A12:  for j being Element of N st j >= i holds N.j in a;
       reconsider i as object;
      take i;
      thus i in the carrier of N;
      take a;
      thus thesis by A12;
    end;
    consider f being Function such that
A13: dom f = Z & rng f c= the carrier of N and
A14: for a being object st a in Z holds P[a,f.a] from FUNCT_1:sch 6(A10);
    reconsider z = rng f as finite Subset of [#]N by A8,A13,FINSET_1:8;
    [#]N is directed by WAYBEL_0:def 6;
    then consider k being Element of N such that
    k in [#]N and
A15: k is_>=_than z by WAYBEL_0:1;
    thus N is_eventually_in A
    proof
      take k;
      let i be Element of N;
A16:  Int A c= A by TOPS_1:16;
      assume
A17:  i >= k;
      now
        let a be set;
        assume
A18:    a in Z;
        then
A19:    f.a in z by A13,FUNCT_1:def 3;
        then reconsider j = f.a as Element of N;
A20:     j <= k by A15,A19;
        P[a,f.a] by A14,A18;
        then consider D1 being set such that
A21:        D1 = a &
   for i,j being Element of N st i = f.a & j >= i
    holds N.j in D1;
        thus N.i in a by A17,ORDERS_2:3,A21,A20;
      end;
      then
A22:  N.i in y by A9,SETFAM_1:43;
      y c= union Y by A6,ZFMISC_1:74;
      then N.i in Int A by A22,A4;
      hence thesis by A16;
    end;
  end;
  hence thesis by YELLOW_6:def 15;
end;
