reserve x,y,X for set;

theorem Th8:
  for S being non empty 1-sorted, N being net of S for A being
  Subset of S,N holds N is_eventually_in A
proof
  let S be non empty 1-sorted, N be net of S;
  let A be Subset of S,N;
  consider i being Element of N such that
A1: A = rng the mapping of N|i by Def2;
  take i;
  let j be Element of N;
  assume i <= j;
  then reconsider j9 = j as Element of N|i by WAYBEL_9:def 7;
  N.j = (N|i).j9 by WAYBEL_9:16
    .= (the mapping of N|i).j9;
  hence thesis by A1,FUNCT_2:4;
end;
