
theorem
  for L being non empty 1-sorted, N being non empty NetStr over L
  for X,Y being set st X c= Y holds
  (N is_eventually_in X implies N is_eventually_in Y) &
  (N is_often_in X implies N is_often_in Y)
proof
  let L be non empty 1-sorted, N be non empty NetStr over L;
  let X,Y be set such that
A1: X c= Y;
  hereby
    assume N is_eventually_in X;
    then consider i being Element of N such that
A2: for j being Element of N st i <= j holds N.j in X;
    thus N is_eventually_in Y
    proof
      take i;
      let j be Element of N;
      assume i <= j;
      then N.j in X by A2;
      hence thesis by A1;
    end;
  end;
  assume
A3: 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
A4: i <= j and
A5: N.j in X by A3;
  take j;
  thus thesis by A1,A4,A5;
end;
