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 Th9:
  Partial_Sums(z * seq) = z * Partial_Sums(seq) & Partial_Sums(seq
  *z ) = Partial_Sums(seq) *z
proof
A1: Partial_Sums(seq *z ) = Partial_Sums(seq) *z
  proof
    defpred P[Nat] means
    Partial_Sums(seq*z).$1= (Partial_Sums(seq)*z).$1;
A2: now
      let n;
      assume
A3:   P[n];
      Partial_Sums( seq*z).(n+1) =Partial_Sums( seq*z).n + (seq*z).(n+1)
      by BHSP_4:def 1
        .=(Partial_Sums(seq).n *z )+ (seq*z).(n+1) by A3,LOPBAN_3:def 6
        .=(Partial_Sums(seq).n*z)+ (seq.(n+1)*z) by LOPBAN_3:def 6
        .= ( Partial_Sums(seq).n + seq.(n+1))*z by LOPBAN_3:38
        .= ( Partial_Sums(seq).(n+1))*z by BHSP_4:def 1
        .= (Partial_Sums(seq)*z ).(n+1) by LOPBAN_3:def 6;
      hence P[n+1];
    end;
    Partial_Sums(seq*z).0 =(seq*z).0 by BHSP_4:def 1
      .= seq.0*z by LOPBAN_3:def 6
      .=Partial_Sums(seq).0 *z by BHSP_4:def 1
      .=(Partial_Sums(seq)*z).0 by LOPBAN_3:def 6;
    then
A4: P[0];
    for n holds P[n] from NAT_1:sch 2(A4,A2);
    then for n being Element of NAT holds P[n];
    hence thesis by FUNCT_2:63;
  end;
  Partial_Sums(z * seq) = z * Partial_Sums(seq)
  proof
    defpred P[Nat] means
    Partial_Sums(z *seq).$1= (z * Partial_Sums(seq)).$1;
A5: now
      let n;
      assume
A6:   P[n];
      Partial_Sums(z * seq).(n+1) =Partial_Sums(z * seq).n + (z * seq).(n+
      1) by BHSP_4:def 1
        .=(z * Partial_Sums(seq).n )+ (z * seq).(n+1) by A6,LOPBAN_3:def 5
        .=(z * Partial_Sums(seq).n )+ (z * seq.(n+1)) by LOPBAN_3:def 5
        .= z * ( Partial_Sums(seq).n + seq.(n+1)) by LOPBAN_3:38
        .= z * ( Partial_Sums(seq).(n+1)) by BHSP_4:def 1
        .= (z * Partial_Sums(seq)).(n+1) by LOPBAN_3:def 5;
      hence P[n+1];
    end;
    Partial_Sums(z*seq).0 =(z * seq).0 by BHSP_4:def 1
      .=z * seq.0 by LOPBAN_3:def 5
      .=z * Partial_Sums(seq).0 by BHSP_4:def 1
      .=(z * Partial_Sums(seq)).0 by LOPBAN_3:def 5;
    then
A7: P[0];
    for n holds P[n] from NAT_1:sch 2(A7,A5);
    then for n being Element of NAT holds P[n];
    hence thesis by FUNCT_2:63;
  end;
  hence thesis by A1;
end;
