
theorem Th11:
  for S, T being LATTICE, f being Function of S, T holds
  (for N being net of S holds f.(lim_inf N) <= lim_inf (f*N)) implies
  f is monotone
proof
  let S, T be LATTICE;
  let f be Function of S, T;
  assume
A1: for N be net of S holds f.(lim_inf N) <= lim_inf (f*N);
  now
    let a, b be Element of S;
    assume
A2: a <= b;
    set N = Net-Str (a,b);
A3: f.(lim_inf N) = f.a by A2,Lm5;
    lim_inf (f*N) = f.a "/\" f.b by Th8;
    then
A4: f.a <= f.a "/\" f.b by A1,A3;
    f.a >= f.a "/\" f.b by YELLOW_0:23;
    then f.a = f.a "/\" f.b by A4,ORDERS_2:2;
    hence f.a <= f.b by YELLOW_0:25;
  end;
  hence thesis;
end;
