
theorem
  for L being continuous complete LATTICE holds lim_inf-Convergence L is
  (DIVERGENCE)
proof
  let L be continuous complete LATTICE;
  let N be net of L, x be Element of L;
  assume that
A1: N in NetUniv L and
A2: not [N,x] in lim_inf-Convergence L;
A3: ex N1 being strict net of L st N1=N & the carrier of N1 in
  the_universe_of the carrier of L by A1,YELLOW_6:def 11;
  not for M being subnet of N holds x = lim_inf M by A1,A2,Def3;
  then
A4: not (x=lim_inf N & for p being greater_or_equal_to_id Function of N,N
  holds x >= inf (N * p)) by Th14;
A5: lim_inf-Convergence L c= [:NetUniv L,the carrier of L:] by YELLOW_6:def 18;
  per cases by A1,A4,Th10;
  suppose
A6: not x=lim_inf N & x<=lim_inf N;
    reconsider N9=N as subnet of N by YELLOW_6:14;
    take N9;
    thus N9 in NetUniv L by A1;
    given M2 being subnet of N9 such that
A7: [M2,x] in lim_inf-Convergence L;
A8: lim_inf N <= lim_inf M2 by WAYBEL21:37;
A9: M2 is subnet of M2 by YELLOW_6:14;
    M2 in NetUniv L by A5,A7,ZFMISC_1:87;
    then lim_inf M2 =x by A7,A9,Def3;
    hence contradiction by A6,A8,YELLOW_0:def 3;
  end;
  suppose
    not x=lim_inf N & not x<=lim_inf N;
    then not x is_S-limit_of N;
    then not [N,x] in Scott-Convergence L by A1,A3,WAYBEL11:def 8;
    then consider M being subnet of N such that
A10: M in NetUniv L and
A11: not ex M1 being subnet of M st [M1,x] in Scott-Convergence L by A1,
YELLOW_6:def 22;
    take M;
    thus M in NetUniv L by A10;
    for M1 being subnet of M holds not [M1,x] in lim_inf-Convergence L
    proof
      let M1 be subnet of M;
A12:  not [M1,x] in Scott-Convergence L by A11;
      assume
A13:  [M1,x] in lim_inf-Convergence L;
      then
A14:  M1 in NetUniv L by A5,ZFMISC_1:87;
      then ex M11 being strict net of L st M11=M1 & the carrier of M11 in
      the_universe_of the carrier of L by YELLOW_6:def 11;
      then
A15:  not x is_S-limit_of M1 by A14,A12,WAYBEL11:def 8;
      M1 is subnet of M1 by YELLOW_6:14;
      then x = lim_inf M1 by A13,A14,Def3;
      hence contradiction by A15;
    end;
    hence thesis;
  end;
  suppose
    not (for M being subnet of N st M in NetUniv L holds x >= inf M);
    then consider M being subnet of N such that
A16: M in NetUniv L and
A17: not x >= inf M;
    take M;
    thus M in NetUniv L by A16;
    for M1 being subnet of M holds not [M1,x] in lim_inf-Convergence L
    proof
      let M1 be subnet of M;
A18:  M1 is subnet of M1 by YELLOW_6:14;
A19:  lim_inf M1 >= lim_inf M & lim_inf M >= inf M by Th1,WAYBEL21:37;
      assume
A20:  [M1,x] in lim_inf-Convergence L;
      then M1 in NetUniv L by A5,ZFMISC_1:87;
      then x = lim_inf M1 by A20,A18,Def3;
      hence contradiction by A17,A19,YELLOW_0:def 2;
    end;
    hence thesis;
  end;
end;
