reserve R for non empty RelStr,
  N for net of R,
  i for Element of N;

theorem Th21:
  for R being non empty transitive RelStr,
  S being non empty set, f be Function of S, the carrier of R
  st S c= the carrier of R & Net-Str(S,f) is directed
  holds Net-Str(S,f) in NetUniv R
proof
  let R be non empty transitive RelStr,
  S be non empty set, f be Function of S, the carrier of R such that
A1: S c= the carrier of R and
A2: Net-Str(S,f) is directed;
  reconsider N = Net-Str(S,f) as strict net of R by A2;
  set UN = the_universe_of the carrier of R;
  reconsider UN as universal set;
  the_transitive-closure_of the carrier of R in UN by CLASSES1:2;
  then the carrier of R in UN by CLASSES1:3,52;
  then
A3: S in UN by A1,CLASSES1:def 1;
  the carrier of N = S by Def10;
  hence thesis by A3,YELLOW_6:def 11;
end;
