
theorem
  for L being non empty RelStr holds lim_inf-Convergence L is (SUBNETS)
proof
  let L be non empty RelStr;
  let N be net of L, M be subnet of N;
  assume
A1: M in NetUniv L;
  let x be Element of L;
  assume
A2: [N,x] in lim_inf-Convergence L;
  lim_inf-Convergence L c= [:NetUniv L,the carrier of L:] by YELLOW_6:def 18;
  then
A3: N in NetUniv L by A2,ZFMISC_1:87;
  for M1 being subnet of M holds x = lim_inf M1
  proof
    let M1 be subnet of M;
    reconsider M19=M1 as subnet of N by YELLOW_6:15;
    x = lim_inf M19 by A2,A3,Def3;
    hence thesis;
  end;
  hence thesis by A1,Def3;
end;
