reserve X for non empty UNITSTR;
reserve a, b for Real;
reserve x, y for Point of X;
reserve X for RealUnitarySpace;
reserve x, y, z, u, v for Point of X;

theorem
  x, y are_orthogonal implies x, - y are_orthogonal
proof
  assume x, y are_orthogonal;
  then - x .|. y = - 0;
  then x .|. (-y) = 0 by Th8;
  hence thesis;
end;
