theorem
  (-seq)*Ns = -(seq*Ns) & (|.seq.|)*Ns = |.(seq*Ns).|
proof
  thus (-seq)*Ns = ((-1r)(#)seq)*Ns by COMSEQ_1:11
    .= (-1r)(#)(seq*Ns) by Th3
    .= -(seq*Ns) by COMSEQ_1:11;
  now
    let n be Element of NAT;
    thus ((|.seq.|)*Ns).n = (|.seq.|).(Ns.n) by FUNCT_2:15
      .= |.seq.(Ns.n).| by VALUED_1:18
      .= |.(seq*Ns).n.| by FUNCT_2:15
      .= (|.seq*Ns.|).n by VALUED_1:18;
  end;
  hence thesis by FUNCT_2:63;
end;
