
theorem ThSc4:
  for L being Z_Lattice, v, u, w being Vector of L,
  a, b being Element of INT.Ring
  holds <; a*v + b*u, w ;> = a * <; v, w ;> + b * <; u, w ;> &
  <; v, a*u + b*w ;> = a * <; v, u ;> + b * <; v, w ;>
  proof
    let L be Z_Lattice, v, u, w be Vector of L, a, b be Element of INT.Ring;
    thus <; a*v + b*u, w ;> = <; a*v, w ;> + <; b*u, w ;> by defZLattice
    .= a * <; v, w ;> + <; b*u, w ;> by defZLattice
    .= a * <; v, w ;> + b * <; u, w ;> by defZLattice;
    thus <; v, a*u + b*w ;> = <; v, a*u ;> + <; v, b*w ;> by ThSc2
    .= a * <; v, u ;> + <; v, b*w ;> by ThSc3
    .= a * <; v, u ;> + b * <; v, w ;> by ThSc3;
  end;
