
theorem Th25:
  for T being Hausdorff TopLattice, N being convergent net of T
for f being Function of T, T st f is continuous holds f.(lim N) in Lim (f * N)
proof
  let T be Hausdorff TopLattice, N be convergent net of T, f be Function of T,
  T such that
A1: f is continuous;
  for V being a_neighborhood of f.(lim N) holds f*N is_eventually_in V
  proof
    let V be a_neighborhood of f.(lim N);
A2: dom f = the carrier of T by FUNCT_2:def 1;
    consider O being a_neighborhood of lim N such that
A3: f.:O c= V by A1,BORSUK_1:def 1;
    lim N in Lim N by YELLOW_6:def 17;
    then N is_eventually_in O by YELLOW_6:def 15;
    then consider s0 being Element of N such that
A4: for j being Element of N st s0 <= j holds N.j in O;
A5: the RelStr of f*N = the RelStr of N by Def8;
    then reconsider s1 = s0 as Element of f*N;
    take s1;
    let j1 be Element of f*N such that
A6: s1 <= j1;
    reconsider j = j1 as Element of N by A5;
    N.j in O by A4,A5,A6,YELLOW_0:1;
    then
A7: f.(N.j) in f.:O by A2,FUNCT_1:def 6;
A8: the carrier of f*N = dom the mapping of N by A5,FUNCT_2:def 1;
    (f*N).j1 = (f * the mapping of N).j1 by Def8
      .= f.(N.j) by A8,FUNCT_1:13;
    hence thesis by A3,A7;
  end;
  hence thesis by YELLOW_6:def 15;
end;
