
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 ThSc5
    .= <; v, v ;> - <; v, u ;> - <; u, v-u ;> by ThSc5
    .= <; v, v ;> - 0 - (<; u, v ;> - <; u, u ;>) by A1,ThSc5
    .= <; v, v ;> - (0 - <; u, u ;>) by A1,defZLattice
    .= ||. v .|| + ||. u .||;
  end;
