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

theorem Th22:
  for S being non empty 1-sorted, N being net of S, X st N
  is_often_in X holds N"X is subnet of N
proof
  let S be non empty 1-sorted, N be net of S, X;
  assume
A1: N is_often_in X;
  then reconsider M = N"X as net of S by Th21;
  M is subnet of N
  proof
    set f = id M;
    the carrier of M c= the carrier of N by Th10;
    then reconsider f as Function of M,N by FUNCT_2:7;
    take f;
    the mapping of M = (the mapping of N)|the carrier of M by Def6;
    hence the mapping of M = (the mapping of N)*f by RELAT_1:65;
    let m be Element of N;
    consider j being Element of N such that
A2: m <= j and
A3: N.j in X by A1;
    j in (the mapping of N)"X by A3,FUNCT_2:38;
    then reconsider n = j as Element of M by Def10;
    take n;
    let p be Element of M such that
A4: n <= p;
    j <= f.p by A4,Th11;
    hence thesis by A2,YELLOW_0:def 2;
  end;
  hence thesis;
end;
