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 Th87:
  (-seq) ^\k = -(seq ^\k)
proof
  now
    let n be Element of NAT;
    thus ((-seq) ^\k).n = (-seq).(n + k) by NAT_1:def 3
      .= -(seq.(n + k)) by BHSP_1:44
      .= -((seq ^\ k).n) by NAT_1:def 3
      .= (-(seq ^\k)).n by BHSP_1:44;
  end;
  hence thesis by FUNCT_2:63;
end;
