
theorem Th24:
  for S,T being non empty TopSpace, N being net of S
  for f being Function of S,T st f is continuous holds f.:Lim N c= Lim (f*N)
proof
  let S,T be non empty TopSpace, N be net of S;
A1: [#]T <> {};
  let f be Function of S,T such that
A2: f is continuous;
  let p be object;
  assume
A3: p in f.:Lim N;
  then reconsider p as Point of T;
  consider x being object such that
A4: x in the carrier of S and
A5: x in Lim N and
A6: p = f.x by A3,FUNCT_2:64;
  reconsider x as Element of S by A4;
  now
    let V be a_neighborhood of p;
A7: p in Int V by CONNSP_2:def 1;
A8: x in f"Int V by A6,A7,FUNCT_2:38;
    f"Int V is open by A1,A2,TOPS_2:43;
    then f"Int V is a_neighborhood of x by A8,CONNSP_2:3;
    then N is_eventually_in f"Int V by A5,YELLOW_6:def 15;
    then consider i being Element of N such that
A9: for j being Element of N st j >= i holds N.j in
    f"Int V;
A10: the mapping of f*N = f*the mapping of N by WAYBEL_9:def 8;
A11: the RelStr of f*N = the RelStr of N by WAYBEL_9:def 8;
    then reconsider i9 = i as Element of f*N;
    thus f*N is_eventually_in V
    proof
      take i9;
      let j9 be Element of f*N;
      reconsider j = j9 as Element of N by A11;
A12:  f.(N.j) = (f*N).j9 by A10,FUNCT_2:15;
      assume j9 >= i9;
      then N.j in f"Int V by A9,A11,YELLOW_0:1;
      then
A13:  f.(N.j) in Int V by FUNCT_2:38;
      Int V c= V by TOPS_1:16;
      hence thesis by A12,A13;
    end;
  end;
  hence thesis by YELLOW_6:def 15;
end;
