
theorem Th1:
  for L being complete LATTICE, N being net of L holds inf N <= lim_inf N
proof
  let L be complete LATTICE, N be net of L;
  set X = the set of all
"/\"({N.i where i is Element of N :i >= j1},L) where j1 is Element
  of N;
  X c= the carrier of L
  proof
    let x be object;
    assume x in X;
    then ex j1 being Element of N st x = "/\"({N.i where i is Element of N :i
    >= j1},L);
    hence thesis;
  end;
  then reconsider X as Subset of L;
  set j = the Element of N;
A1: {N.i where i is Element of N:i >= j} c= rng the mapping of N
  proof
    let x be object;
A2: dom(the mapping of N) = the carrier of N by FUNCT_2:def 1;
    assume x in {N.i where i is Element of N:i >= j};
    then consider i being Element of N such that
A3: x = N.i and
    i >= j;
    x = (the mapping of N).i by A3,WAYBEL_0:def 8;
    hence thesis by A2,FUNCT_1:def 3;
  end;
  reconsider X as Subset of L;
  set x = "/\"({N.i where i is Element of N:i >= j},L);
  ex_inf_of {N.i where i is Element of N:i >= j},L & ex_inf_of rng the
  mapping of N,L by YELLOW_0:17;
  then
  "/\"({N.i where i is Element of N:i >= j},L) >= "/\"(rng the mapping of
  N,L) by A1,YELLOW_0:35;
  then x >= Inf(the mapping of N) by YELLOW_2:def 6;
  then
A4: inf N <= x by WAYBEL_9:def 2;
  ex_sup_of X,L by YELLOW_0:17;
  then x in X & X is_<=_than "\/"(X,L) by YELLOW_0:def 9;
  then x <= "\/"(X,L) by LATTICE3:def 9;
  hence thesis by A4,YELLOW_0:def 2;
end;
