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 Th26:
  for z,w st z,w are_commutative holds Partial_Sums(z rExpSeq).k *
Partial_Sums(w rExpSeq).k -Partial_Sums((z+w) rExpSeq).k = Partial_Sums(Conj(k,
  z,w)).k
proof
  let z,w;
  assume z,w are_commutative;
  then
A1: (Partial_Sums(z rExpSeq).k) * (Partial_Sums(w rExpSeq).k) -Partial_Sums(
  (z+w) rExpSeq).k =Partial_Sums(z rExpSeq).k * Partial_Sums(w rExpSeq).k -
  Partial_Sums(Alfa(k,z,w)).k by Th25
    .=( Partial_Sums(z rExpSeq)* (Partial_Sums(w rExpSeq).k) ).k -
  Partial_Sums(Alfa(k,z,w)).k by LOPBAN_3:def 6
    .=(Partial_Sums((z rExpSeq)*(Partial_Sums(w rExpSeq).k))) .k -
  Partial_Sums(Alfa(k,z,w)).k by Th9
    .=(Partial_Sums( (z rExpSeq)*(Partial_Sums(w rExpSeq).k) ) -Partial_Sums
  (Alfa(k,z,w))).k by NORMSP_1:def 3
    .=Partial_Sums((( (z rExpSeq)*(Partial_Sums(w rExpSeq).k))) -(Alfa(k,z,w
  ))).k by LOPBAN_3:16;
  for l be Nat st l <= k holds ( (z rExpSeq)*(Partial_Sums(w
  rExpSeq).k) - (Alfa(k,z,w))).l =Conj(k,z,w).l
  proof
    let l be Nat such that
A2: l <= k;
    thus ( (z rExpSeq)*( Partial_Sums(w rExpSeq).k) - (Alfa(k,z,w)) ).l = ((z
    rExpSeq)*(Partial_Sums(w rExpSeq).k) ).l -(Alfa(k,z,w)).l by NORMSP_1:def 3
      .= ( (z rExpSeq).l )*( Partial_Sums(w rExpSeq).k) -Alfa(k,z,w).l by
LOPBAN_3:def 6
      .=((z rExpSeq).l) * ( Partial_Sums(w rExpSeq).k) -((z rExpSeq).l) * (
    Partial_Sums(w rExpSeq).(k-'l)) by A2,Def8
      .=((z rExpSeq).l) * ( Partial_Sums(w rExpSeq).k -Partial_Sums(w
    rExpSeq).(k-'l) ) by LOPBAN_3:38
      .=Conj(k,z,w).l by A2,Def9;
  end;
  hence thesis by A1,Th11;
end;
