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

theorem Th23:
  for S being non empty 1-sorted, N being net of S, X for M being
  subnet of N st M = N"X holds M is_eventually_in X
proof
  let S be non empty 1-sorted, N be net of S, X;
  let M be subnet of N such that
A1: M = N"X;
  set i = the Element of M;
  take i;
  let j be Element of M such that
  i <= j;
  j in the carrier of M;
  then j in (the mapping of N)"X by A1,Def10;
  then
A2: (the mapping of N).j in X by FUNCT_1:def 7;
  the mapping of M = (the mapping of N)|the carrier of M by A1,Def6;
  hence thesis by A2,FUNCT_1:49;
end;
