reserve x,y,z,X for set,
  T for Universe;

theorem Th21:
  for S being non empty 1-sorted, N be net of S, X st N
  is_often_in X holds N"X is non empty directed
proof
  let S be non empty 1-sorted, N be net of S, X such that
A1: N is_often_in X;
  set i = the Element of N;
  consider j being Element of N such that
  i <= j and
A2: N.j in X by A1;
A3: j in (the mapping of N)"X by A2,FUNCT_2:38;
  hence the carrier of N"X is non empty by Def10;
  reconsider M = N"X as non empty full SubNetStr of N by A3,Def10;
  M is directed
  proof
    let i,j be Element of M;
    the carrier of M c= the carrier of N by Th10;
    then reconsider x = i, y = j as Element of N;
    consider z being Element of N such that
A4: x <= z & y <= z by Def3;
    consider e being Element of N such that
A5: z <= e and
A6: N.e in X by A1;
    e in (the mapping of N)"X by A6,FUNCT_2:38;
    then reconsider k = e as Element of M by Def10;
    take k;
    x <= e & y <= e by A4,A5,YELLOW_0:def 2;
    hence thesis by Th12;
  end;
  hence thesis;
end;
