theorem
  (a + b) * seq = a * seq + b * seq
proof
  let n be Element of NAT;
  thus ((a + b) * seq).n = (a + b) * seq.n by NORMSP_1:def 5
    .= a * seq.n + b * seq.n by RLVECT_1:def 6
    .= (a * seq).n + b * seq.n by NORMSP_1:def 5
    .= (a * seq).n + (b * seq).n by NORMSP_1:def 5
    .= (a * seq + b * seq).n by NORMSP_1:def 2;
end;
