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 Th11:
  (for n st n <= m holds seq1.n = seq2.n) implies Partial_Sums(
  seq1).m =Partial_Sums(seq2).m
proof
  defpred P[Nat] means
   $1 <= m implies Partial_Sums(seq1).$1=Partial_Sums(seq2).$1;
  assume
A1: for n st n <= m holds seq1.n = seq2.n;
A2: for k be Nat st P[k] holds P[k+1]
  proof
    let k be Nat such that
A3: P[k];
    assume
A4: k+1 <= m;
    k < k+1 by XREAL_1:29;
    hence Partial_Sums(seq1).(k+1) =Partial_Sums(seq2).k+seq1.(k+1) by A3,A4,
BHSP_4:def 1,XXREAL_0:2
      .=Partial_Sums(seq2).k+seq2.(k+1) by A1,A4
      .=Partial_Sums(seq2).(k+1) by BHSP_4:def 1;
  end;
A5: P[0]
  proof
    assume 0 <= m;
    thus Partial_Sums(seq1).0=seq1.0 by BHSP_4:def 1
      .=seq2.0 by A1
      .=Partial_Sums(seq2).0 by BHSP_4:def 1;
  end;
  for k holds P[k] from NAT_1:sch 2(A5,A2);
  hence thesis;
end;
