reserve X for Complex_Banach_Algebra,
  w,z,z1,z2 for Element of X,
  k,l,m,n,n1, n2 for Nat,
  seq,seq1,seq2,s,s9 for sequence of X,
  rseq for Real_Sequence;

theorem Th26:
  0 <= (||. z .|| rExpSeq).n
proof
  defpred P[Nat] means 0 <= ||.z.|| |^$1;
A1: for k st P[k] holds P[k+1]
  proof
    let k such that
A2: P[k];
A3: 0 <= ||.z.|| by CLVECT_1:105;
    ||.z.|| |^ (k+1) = ||.z.|| |^ k * ||.z.|| by NEWTON:6;
    hence thesis by A2,A3;
  end;
A4: P[0] by NEWTON:4;
  for k holds P[k] from NAT_1:sch 2(A4,A1);
  then
A5: 0 <= ||.z.|| |^n;
A6: ||.z.|| |^n/(n!) = ||.z.|| |^n *((n!))" by XCMPLX_0:def 9;
  thus thesis by A5,A6,SIN_COS:def 5;
end;
