
theorem ThSc1:
  for L being Z_Lattice, v, u being Vector of L holds
  <; v, -u ;> = - <; v, u ;> & <; -v, u ;> = - <; v, u ;>
  proof
    let L be Z_Lattice, v, u be Vector of L;
    thus <; v, -u ;> = <; v, (-1.(INT.Ring))*u ;> by ZMODUL01:2
    .= <; (-1.(INT.Ring))*u, v ;> by defZLattice
    .= (-1) * <; u, v ;> by defZLattice
    .= - <; u, v ;>
    .= - <; v, u ;> by defZLattice;
    thus <; -v, u ;> = <; (-1.(INT.Ring))*v, u ;> by ZMODUL01:2
    .= (-1.(INT.Ring)) * <; v, u ;> by defZLattice
    .= - <; v, u ;>;
  end;
