
theorem
  for p be complex-valued FinSequence
  for x be Complex holds
  |.p^<*x*>.| = |.p.|^<*|.x.|*> & |.<*x*>^p.| = <*|.x.|*>^|.p.|
proof
  let p be complex-valued FinSequence;
  let x be Complex;
  thus |.p^<*x*>.| = |.p.|^|.<*x*>.| by Th12
    .= |.p.|^<*|.x.|*> by Th9;
  thus |.<*x*>^p.| = |.<*x*>.|^|.p.| by Th12
    .= <*|.x.|*>^|.p.| by Th9;
end;
