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 Th75:
  seq is bounded implies -seq is bounded
proof
  assume seq is bounded;
  then consider M such that
A1: M > 0 and
A2: for n holds ||.seq.n.|| <= M;
  now
    let n;
    ||.(- seq).n.|| = ||.- (seq.n).|| by BHSP_1:44
      .= ||.seq.n.|| by CSSPACE:47;
    hence ||.(- seq).n.|| <= M by A2;
  end;
  hence thesis by A1;
end;
