
theorem
  for L being Z_Lattice, v, u being Vector of L holds
  v, u are_orthogonal implies v, -u are_orthogonal & -v, u are_orthogonal &
  -v, -u are_orthogonal
  proof
    let L be Z_Lattice, v, u be Vector of L;
    assume A1: v, u are_orthogonal;
     <; v, -u ;> = - <; v, u ;> by ThSc1
    .= 0 by A1;
    hence v, -u are_orthogonal;
    A2: <; -v, u ;> = - <; v, u ;> by ThSc1
    .= 0 by A1;
    hence -v, u are_orthogonal;
    <; -v, -u ;> = - <; -v, u ;> by ThSc1
    .= 0 by A2;
    hence -v, -u are_orthogonal;
  end;
