
theorem ::3.1 Proposition (1)=>(2)
  for L being complete LATTICE, N being net of L, x being Element of L
holds (for M being subnet of N holds x = lim_inf M) implies (x=lim_inf N & for
  M being subnet of N holds x >= inf M)
proof
  let L be complete LATTICE, N be net of L, x be Element of L;
  assume
A1: for M being subnet of N holds x = lim_inf M;
  N is subnet of N by YELLOW_6:14;
  hence x=lim_inf N by A1;
  let M be subnet of N;
  x = lim_inf M by A1;
  hence thesis by Th1;
end;
