
theorem
  for L being non empty 1-sorted, N being non empty NetStr over L
  for X being set holds
  N is_eventually_in X iff not N is_often_in (the carrier of L) \ X
proof
  let L be non empty 1-sorted, N be non empty NetStr over L, X be set;
  set Y = (the carrier of L) \ X;
  thus N is_eventually_in X implies not N is_often_in Y
  proof
    given i being Element of N such that
A1: for j being Element of N st i <= j holds N.j in X;
    take i;
    let j be Element of N;
    assume i <= j;
    then N.j in X by A1;
    hence thesis by XBOOLE_0:def 5;
  end;
  given i being Element of N such that
A2: for j being Element of N st i <= j holds not N.j in Y;
  take i;
  let j be Element of N;
  assume i <= j;
  then not N.j in Y by A2;
  hence thesis by XBOOLE_0:def 5;
end;
