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

theorem Th27:
  for S being non empty 1-sorted, e being Element of S, X being set holds
  Net-Str e is_eventually_in X iff e in X
proof
  let S be non empty 1-sorted, e be Element of S, X be set;
  set N = Net-Str e;
  the carrier of N = {e} by Def11;
  then reconsider d = e as Element of N by TARSKI:def 1;
  hereby
    assume N is_eventually_in X;
    then consider x being Element of N such that
A1: for y being Element of N st x <= y holds N.y in X;
    N.x in X by A1;
    hence e in X by Th26;
  end;
  assume
A2: e in X;
  take d;
  let j be Element of N such that d <= j;
  thus thesis by A2,Th26;
end;
