reserve X for RealUnitarySpace,
  x, g, g1, h for Point of X,
  a, p, r, M, M1, M2 for Real,
  seq, seq1, seq2, seq3 for sequence of X,
  Nseq for increasing sequence of NAT,

  k, l, l1, l2, l3, n, m, m1, m2 for Nat;

theorem Th13:
  (seq1 + seq2) ^\k = (seq1 ^\k) + (seq2 ^\k)
proof
  now
    let n be Element of NAT;
    thus ((seq1 + seq2) ^\k).n = (seq1 + seq2).(n + k) by NAT_1:def 3
      .= seq1.(n + k) + seq2.(n + k) by NORMSP_1:def 2
      .= (seq1 ^\k).n + seq2.(n + k) by NAT_1:def 3
      .= (seq1 ^\k).n + (seq2 ^\k).n by NAT_1:def 3
      .= ((seq1 ^\k) + (seq2 ^\k)).n by NORMSP_1:def 2;
  end;
  hence thesis by FUNCT_2:63;
end;
