reserve q,th,r for Real,
  a,b,p for Real,
  w,z for Complex,
  k,l,m,n,n1,n2 for Nat,
  seq,seq1,seq2,cq1 for Complex_Sequence,
  rseq,rseq1,rseq2 for Real_Sequence,
  rr for set,
  hy1 for 0-convergent non-zero Real_Sequence;

theorem Th15:
  Partial_Sums(z ExpSeq).k * Partial_Sums(w ExpSeq).k
  -Partial_Sums((z+w) ExpSeq).k = Partial_Sums(Conj(k,z,w)).k
proof
A1:  k in NAT by ORDINAL1:def 12;
A2: (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 Th14
    .= ((Partial_Sums(w ExpSeq).k) (#) Partial_Sums(z ExpSeq)).k
  -Partial_Sums(Alfa(k,z,w)).k by VALUED_1:6
    .=Partial_Sums((Partial_Sums(w ExpSeq).k) (#) (z ExpSeq)).k
  -Partial_Sums(Alfa(k,z,w)).k by COMSEQ_3:29
    .=Partial_Sums((Partial_Sums(w ExpSeq).k) (#) (z ExpSeq)).k
  +(-Partial_Sums(Alfa(k,z,w))).k by VALUED_1:8
    .=(Partial_Sums((Partial_Sums(w ExpSeq).k) (#) (z ExpSeq))
  -Partial_Sums(Alfa(k,z,w))).k by VALUED_1:1,A1
    .=Partial_Sums((((Partial_Sums(w ExpSeq).k) (#) (z ExpSeq)))
  -(Alfa(k,z,w))).k by COMSEQ_3:28;
 for l be Nat st l <= k holds
  ((Partial_Sums(w ExpSeq).k) (#) (z ExpSeq) - (Alfa(k,z,w))).l =Conj(k,z,w).l
  proof
    let l be Nat such that
A3: l <= k;
A4: l in NAT by ORDINAL1:def 12;
    thus ( ( Partial_Sums(w ExpSeq).k) (#) (z ExpSeq) - (Alfa(k,z,w)) ).l
    =(( Partial_Sums(w ExpSeq).k) (#) (z ExpSeq)).l
    +(-Alfa(k,z,w)).l by VALUED_1:1,A4
      .=(( Partial_Sums(w ExpSeq).k) (#) (z ExpSeq)).l -Alfa(k,z,w).l
    by VALUED_1:8
      .=(( Partial_Sums(w ExpSeq).k ) * ( (z ExpSeq).l ))
    -Alfa(k,z,w).l by VALUED_1:6
      .=((z ExpSeq).l) * ( Partial_Sums(w ExpSeq).k)
    -((z ExpSeq).l) * ( Partial_Sums(w ExpSeq).(k-'l)) by A3,Def11
      .=((z ExpSeq).l) * ( Partial_Sums(w ExpSeq).k
    -Partial_Sums(w ExpSeq).(k-'l) )
      .=Conj(k,z,w).l by A3,Def13;
  end;
  hence thesis by A2,COMSEQ_3:35;
end;
