reserve x,y,z,X for set,
  T for Universe;

theorem Th17:
  for L being non empty 1-sorted, N being net of L, X,Y being set
  st N is_eventually_in X & N is_eventually_in Y holds X meets Y
proof
  let L be non empty 1-sorted, N be net of L, X,Y be set;
  assume N is_eventually_in X;
  then consider i1 being Element of N such that
A1: for j being Element of N st i1 <= j holds N.j in X;
  assume N is_eventually_in Y;
  then consider i2 being Element of N such that
A2: for j being Element of N st i2 <= j holds N.j in Y;
  consider i being Element of N such that
A3: i1 <= i and
A4: i2 <= i by Def3;
A5: N.i in Y by A2,A4;
  N.i in X by A1,A3;
  hence thesis by A5,XBOOLE_0:3;
end;
