
theorem ThSc5:
  for L being Z_Lattice, v, u, w being Vector of L holds
  <; v-u, w ;> = <; v, w ;> - <; u, w ;> &
  <; v, u-w ;> = <; v, u ;> - <; v, w ;>
  proof
    let L be Z_Lattice, v, u, w be Vector of L;
    thus <; v-u, w ;> = <; v+(-1.(INT.Ring))*u, w ;> by ZMODUL01:2
    .= <; v, w ;> + <; (-1.(INT.Ring))*u, w ;> by defZLattice
    .= <; v, w ;> + (-1.(INT.Ring)) * <; u, w ;> by defZLattice
    .= <; v, w ;> - <; u, w ;>;
    thus <; v, u-w ;> = <; v, u+(-1.(INT.Ring))*w ;> by ZMODUL01:2
    .= <; v, u ;> + <; v, (-1.(INT.Ring))*w ;> by ThSc2
    .= <; v, u ;> + (-1.(INT.Ring)) * <; v, w ;> by ThSc3
    .= <; v, u ;> - <; v, w ;>;
  end;
