
theorem Th22:
  for L being non empty reflexive RelStr holds lim_inf-Convergence
  L c= Scott-Convergence L
proof
  let L be non empty reflexive RelStr;
  let Ns,xs be object;
  assume
A1: [Ns,xs] in lim_inf-Convergence L;
A2: lim_inf-Convergence L c= [:NetUniv L,the carrier of L:] by YELLOW_6:def 18;
  then reconsider x=xs as Element of L by A1,ZFMISC_1:87;
  Ns in NetUniv L by A2,A1,ZFMISC_1:87;
  then consider N being strict net of L such that
A3: N = Ns and
  the carrier of N in the_universe_of the carrier of L by YELLOW_6:def 11;
A4: N in NetUniv L by A2,A1,A3,ZFMISC_1:87;
  N is subnet of N by YELLOW_6:14;
  then x = lim_inf N by A1,A3,A4,Def3;
  then x is_S-limit_of N;
  hence thesis by A3,A4,WAYBEL11:def 8;
end;
