reserve X for ComplexUnitarySpace;
reserve x, y, w, g, g1, g2 for Point of X;
reserve z for Complex;
reserve p, q, r, M, M1, M2 for Real;
reserve seq, seq1, seq2, seq3 for sequence of X;
reserve k,n,m for Nat;
reserve Nseq for increasing sequence of NAT;

theorem Th86:
  (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;
