
theorem Th3:
  for L being complete LATTICE, N being net of L, x being Element
of L st N in NetUniv L holds (for M being subnet of N st M in NetUniv L holds x
  = lim_inf M) implies (x=lim_inf N & for M being subnet of N st M in NetUniv L
  holds x >= inf M)
proof
  let L be complete LATTICE, N be net of L, x be Element of L;
  assume
A1: N in NetUniv L;
  assume
A2: for M being subnet of N st M in NetUniv L holds x = lim_inf M;
  N is subnet of N by YELLOW_6:14;
  hence x=lim_inf N by A1,A2;
  let M be subnet of N;
  assume M in NetUniv L;
  then x = lim_inf M by A2;
  hence thesis by Th1;
end;
