
theorem
  for L being complete LATTICE, N being net of L, x being Element of L
  st N in NetUniv L holds [N,x] in lim_inf-Convergence L iff for M being subnet
  of N st M in NetUniv L holds x = lim_inf M
proof
  let L be complete LATTICE;
  let N be net of L;
  let x be Element of L;
  assume
A1: N in NetUniv L;
  hence [N,x] in lim_inf-Convergence L implies for M being subnet of N st M in
  NetUniv L holds x = lim_inf M by Def3;
  assume
A2: for M being subnet of N st M in NetUniv L holds x = lim_inf M;
  then for M being subnet of N st M in NetUniv L holds x >= inf M by A1,Th3;
  then
A3: for p being greater_or_equal_to_id Function of N,N holds x >= inf (N * p
  ) by A1,Th10;
  x=lim_inf N by A1,A2,Th3;
  then for M being subnet of N holds x = lim_inf M by A3,Th14;
  hence thesis by A1,Def3;
end;
