reserve x for set;

theorem Th4:
  for L1, L2 being /\-complete up-complete Semilattice st the
  RelStr of L1 = the RelStr of L2 for N1 being net of L1, N2 being net of L2 st
  the RelStr of N1 = the RelStr of N2 & the mapping of N1 = the mapping of N2
  holds lim_inf N1 = lim_inf N2
proof
  let L1, L2 be /\-complete up-complete Semilattice such that
A1: the RelStr of L1 = the RelStr of L2;
  let N1 be net of L1, N2 be net of L2 such that
A2: the RelStr of N1 = the RelStr of N2 & the mapping of N1 = the
  mapping of N2;
  deffunc I2(Element of N2) = {N2.i where i is Element of N2: i >= $1};
  deffunc I1(Element of N1) = {N1.i where i is Element of N1: i >= $1};
  set U1 = the set of all "/\" (I1(j), L1) where j is Element of N1;
  set U2 = the set of all "/\" (I2(j), L2) where j is Element of N2;
A3: lim_inf N1 = "\/"(U1, L1) & lim_inf N2 = "\/"(U2, L2) by WAYBEL11:def 6;
  U1 c= the carrier of L1
  proof
    let x be object;
    assume x in U1;
    then ex j being Element of N1 st x = "/\"(I1(j), L1);
    hence thesis;
  end;
  then reconsider U1 as Subset of L1;
  U1 is non empty directed by WAYBEL32:7;
  then
A4: ex_sup_of U1, L1 by WAYBEL_0:75;
  U1 c= U2 & U2 c= U1 by A1,A2,Lm2;
  then U1 = U2;
  hence thesis by A1,A3,A4,YELLOW_0:26;
end;
