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 Th25:
  for z,w st z,w are_commutative holds Partial_Sums(z ExpSeq).k *
Partial_Sums(w ExpSeq).k - Partial_Sums((z+w) ExpSeq).k = Partial_Sums(Conj(k,z
  ,w)).k
proof
  let z,w;
  assume z,w are_commutative;
  then
A1: (Partial_Sums(z ExpSeq).k) * (Partial_Sums(w ExpSeq).k) -Partial_Sums((z
  +w) ExpSeq).k =Partial_Sums(z ExpSeq).k * Partial_Sums(w ExpSeq).k -
  Partial_Sums(Alfa(k,z,w)).k by Th24
    .=( Partial_Sums(z ExpSeq)* (Partial_Sums(w ExpSeq).k) ).k -Partial_Sums
  (Alfa(k,z,w)).k by LOPBAN_3:def 6
    .=(Partial_Sums((z ExpSeq)*(Partial_Sums(w ExpSeq).k))) .k -Partial_Sums
  (Alfa(k,z,w)).k by Th9
    .=(Partial_Sums( (z ExpSeq)*(Partial_Sums(w ExpSeq).k) ) -Partial_Sums(
  Alfa(k,z,w))).k by NORMSP_1:def 3
    .=Partial_Sums((( (z ExpSeq)*(Partial_Sums(w ExpSeq).k))) -(Alfa(k,z,w))
  ).k by CLOPBAN3:16;
  for l be Nat st l <= k holds ((z ExpSeq)*(Partial_Sums(w
  ExpSeq).k) - (Alfa(k,z,w))).l =Conj(k,z,w).l
  proof
    let l be Nat such that
A2: l <= k;
    thus ((z ExpSeq)*( Partial_Sums(w ExpSeq).k) - (Alfa(k,z,w))).l = ((z
    ExpSeq)*(Partial_Sums(w ExpSeq).k) ).l - (Alfa(k,z,w)).l by NORMSP_1:def 3
      .= ( (z ExpSeq).l )*( Partial_Sums(w ExpSeq).k) - Alfa(k,z,w).l by
LOPBAN_3:def 6
      .=((z ExpSeq).l) * ( Partial_Sums(w ExpSeq).k) -((z ExpSeq).l) * (
    Partial_Sums(w ExpSeq).(k-'l)) by A2,Def4
      .=((z ExpSeq).l) * ( Partial_Sums(w ExpSeq).k -Partial_Sums(w ExpSeq).
    (k-'l) ) by CLOPBAN3:38
      .=Conj(k,z,w).l by A2,Def5;
  end;
  hence thesis by A1,Th11;
end;
