reserve x for set;

theorem Th14:
  for L being complete LATTICE, F being proper Filter of
  BoolePoset [#]L holds a_net F in NetUniv L
proof
  let L be complete LATTICE;
  let F be proper Filter of BoolePoset [#]L;
  set S = {[a, f] where a is Element of L, f is Element of F: a in f};
  set UN = the_universe_of the carrier of L;
  reconsider UN as universal set;
  the_transitive-closure_of the carrier of L in UN by CLASSES1:2;
  then
A1: the carrier of L in UN by CLASSES1:3,52;
  then bool the carrier of L in UN by CLASSES2:59;
  then
A2: [:the carrier of L, bool the carrier of L:] in UN by A1,CLASSES2:61;
  S c= [:the carrier of L, bool the carrier of L:] by Lm4;
  then S = the carrier of a_net F & S in UN by A2,CLASSES1:def 1,YELLOW19:def 4
;
  hence thesis by YELLOW_6:def 11;
end;
