
theorem
  for L being non empty 1-sorted, N being non empty NetStr over L
  for X being set holds
  N is_often_in X iff not N is_eventually_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_often_in X implies not N is_eventually_in Y
  proof
    assume
A1: for i being Element of N ex j being Element of N st i <= j & N.j in X;
    let i be Element of N;
    consider j being Element of N such that
A2: i <= j and
A3: N.j in X by A1;
    take j;
    thus thesis by A2,A3,XBOOLE_0:def 5;
  end;
  assume
  A4: for
 i being Element of N ex j being Element of N st i <= j & not N.j in Y;
  let i be Element of N;
  consider j being Element of N such that
A5: i <= j and
A6: not N.j in Y by A4;
  take j;
  thus thesis by A5,A6,XBOOLE_0:def 5;
end;
