
theorem Th11:
  for L being non empty RelStr, N being non empty NetStr over L holds
  N is eventually-directed iff
  for i being Element of N ex j being Element of N st
  for k being Element of N st j <= k holds N.i <= N.k
proof
  let L be non empty RelStr, N be non empty NetStr over L;
A1: now
    let i be Element of N;
    defpred P[Element of L] means N.i <= $1;
    thus N is_eventually_in {N.j where j is Element of N: P[N.j]} iff
    ex k being Element of N st for l being Element of N st k <= l holds P[N.l]
    from HoldsEventually;
  end;
  hereby
    assume
A2: N is eventually-directed;
    let i be Element of N;
    N is_eventually_in {N.j where j is Element of N: N.i <= N.j} by A2;
    hence ex j being Element of N st
    for k being Element of N st j <= k holds N.i <= N.k by A1;
  end;
  assume
A3: for i being Element of N ex j being Element of N st
  for k being Element of N st j <= k holds N.i <= N.k;
  let i be Element of N;
  ex j being Element of N st
  for k being Element of N st j <= k holds N.i <= N.k by A3;
  hence thesis by A1;
end;
