reserve X for 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 Th31:
  |.Partial_Sums(||.Conj(k,z,w).||).n.|=Partial_Sums(||.Conj(k,z, w).||).n
proof
A1: Partial_Sums(||.Conj(k,z,w).||).0=(||.Conj(k,z,w).||).0 by SERIES_1:def 1;
A2: now
    let n be Nat;
    ||.Conj(k,z,w).||.n=||.(Conj(k,z,w)).n.|| by NORMSP_0:def 4;
    hence 0 <= ||.Conj(k,z,w).||.n by NORMSP_1:4;
  end;
  then Partial_Sums(||.Conj(k,z,w).||) is non-decreasing by SERIES_1:16;
  then
A3: Partial_Sums(||.Conj(k,z,w).||).0 <= Partial_Sums(||.Conj(k,z,w).||).n
  by SEQM_3:6;
  0 <= (||.Conj(k,z,w).||).0 by A2;
  hence thesis by A3,A1,ABSVALUE:def 1;
end;
