
theorem
  for L being Z_Lattice, v, u being Vector of L holds
  v, u are_orthogonal implies ||. v + u .|| = ||. v .|| + ||. u .||
  proof
    let L be Z_Lattice, v, u be Vector of L;
    assume A1: v, u are_orthogonal;
    thus ||. v + u .|| = <; v, v+u ;> + <; u, v+u ;> by defZLattice
    .= <; v, v ;> + <; v, u ;> + <; u, v+u ;> by ThSc2
    .= <; v, v ;> + 0 + (<; u, v ;> + <; u, u ;>) by A1,ThSc2
    .= <; v, v ;> + (0 + <; u, u ;>) by A1,defZLattice
    .= ||. v .|| + ||. u .||;
  end;
