
theorem ThRatLat1:
  for L being Z_Lattice, I being Basis of L
  st for v, u being Vector of L st v in I & u in I holds
  <; v, u ;> in RAT
  holds L is RATional
  proof
    let L be Z_Lattice, I be Basis of L such that
    A1: for v, u being Vector of L st v in I & u in I holds
    <; v, u ;> in RAT;
    for v, u being Vector of L holds <; v, u ;> in RAT by A1,ThRatLat1A;
    hence thesis by ZMODLAT2:def 1;
  end;
