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 Th6:
  (z+w) |^ n = Partial_Sums(Expan(n,z,w)).n
proof
A1: 0-'0=0 by XREAL_1:232;
  defpred X[Nat] means
(z+w) |^ $1 = Partial_Sums(Expan($1,z,w)).$1;
 Partial_Sums(Expan(0,z,w)).0 = Expan(0,z,w).0 by SERIES_1:def 1
    .= (Coef(0)).0 * (z |^ 0) * (w |^ 0) by A1,Def9
    .= 1/(1 * 1) * z |^ 0 * w |^ 0 by A1,Def6,Th1
    .= 1r * (w GeoSeq).0 by COMSEQ_3:def 1
    .= 1r by COMSEQ_3:def 1;
then A2: X[0] by COMSEQ_3:def 1;
A3: for n st X[n] holds X[n+1]
  proof
    let n such that
A4: (z+w) |^ n = Partial_Sums(Expan(n,z,w)).n;
A5:  n in NAT by ORDINAL1:def 12;
A6: (z+w) |^ (n+1) =((z+w) GeoSeq ).n * (z+w) by COMSEQ_3:def 1
      .=((z+w) (#) Partial_Sums(Expan(n,z,w))).n by A4,VALUED_1:6
      .=(Partial_Sums((z+w) (#) Expan(n,z,w))).n by COMSEQ_3:29;
 now
      let k be Element of NAT;
      thus
      ((z+w) (#) Expan(n,z,w)).k=(z+w) * Expan(n,z,w).k by VALUED_1:6
        .=z * Expan(n,z,w).k+w * Expan(n,z,w).k
        .=(z (#) Expan(n,z,w)).k+w * Expan(n,z,w).k by VALUED_1:6
        .=(z (#) Expan(n,z,w)).k+(w (#) Expan(n,z,w)).k by VALUED_1:6
        .=((z (#) Expan(n,z,w))+(w (#) Expan(n,z,w))).k by VALUED_1:1;
    end;
then
 (z+w) (#) Expan(n,z,w)=(z (#) Expan(n,z,w))+(w (#) Expan(n,z,w));
    then A7: (
z+w) |^ (n+1) =( Partial_Sums(z (#) (Expan(n,z,w))) +Partial_Sums ((w (#)
    (Expan(n,z,w))))).n by A6,COMSEQ_3:27
      .= Partial_Sums((z (#) Expan(n,z,w))).n
    +Partial_Sums((w (#) Expan(n,z,w))).n by VALUED_1:1,A5;
A8: Partial_Sums((z (#) Expan(n,z,w))).(n+1)
    =Partial_Sums((z (#) Expan(n,z,w))).n+(z (#) Expan(n,z,w)).(n+1)
    by SERIES_1:def 1
      .=Partial_Sums((z (#) Expan(n,z,w))).n
    +z * Expan(n,z,w).(n+1) by VALUED_1:6;
 n < n+1 by XREAL_1:29;
then A9: Expan(n,z,w).(n+1)=0c by Def9;
A10: Partial_Sums((w (#) Expan(n,z,w))).(n+1)
    =Partial_Sums((w (#) Expan(n,z,w))).n+(w (#) Expan(n,z,w)).(n+1)
    by SERIES_1:def 1
      .=Partial_Sums((w (#) Expan(n,z,w))).n
    +w * Expan(n,z,w).(n+1) by VALUED_1:6;
A11: Partial_Sums((z (#) Expan(n,z,w))).(n+1)
    =Partial_Sums(Shift((z (#) Expan(n,z,w)))).(n+1)
    +(z (#) Expan(n,z,w)).(n+1) by Th5;
 0 +n < n+1 by XREAL_1:29;
then  Expan(n,z,w).(n+1)=0c by Def9;
then  z * Expan(n,z,w).(n+1)=0c;
then A12: Partial_Sums((z (#) Expan(n,z,w))).(n+1)
    =Partial_Sums(Shift((z (#) Expan(n,z,w)))).(n+1) +0c by A11,VALUED_1:6
      .= Partial_Sums(Shift((z (#) Expan(n,z,w)))).(n+1);
 now
      let k be Element of NAT;
A13:  now
        assume
A14:    n+1 < k;
A15:    0+1 <= n+1 by XREAL_1:6;
A16:    n+1-1 < k -1 by A14,XREAL_1:9;
then A17:    n+0 < k-1+1 by XREAL_1:8;
        A18:    k-1 = k-'1 by A14,A15,XREAL_1:233,XXREAL_0:2;
    ((w (#) Expan(n,z,w))+Shift((z (#) Expan(n,z,w)))).k
        =(w (#) Expan(n,z,w)).k + (Shift((z (#) Expan(n,z,w)))).k
        by VALUED_1:1
          .=(w * Expan(n,z,w).k) + (Shift((z (#) Expan(n,z,w)))).k
        by VALUED_1:6
          .=(w * Expan(n,z,w).k) + ((z (#) Expan(n,z,w))).(k-'1) by A17,Th4
          .=(w * Expan(n,z,w).k) + (z * (Expan(n,z,w).(k-'1))) by VALUED_1:6
          .=(w * 0c )+ (z * ((Expan(n,z,w).(k-'1)))) by A17,Def9
          .=0c + (z * 0c) by A16,A18,Def9
          .=0c;
        hence ((w (#) Expan(n,z,w))+Shift((z (#) Expan(n,z,w)))).k
        =Expan(n+1,z,w).k by A14,Def9;
      end;
  now
        assume
A19:    k <= (n+1);
A20:    now
          assume
A21:      k=n+1;
A22:      n < n+1 by XREAL_1:29;
A23:      n-'n= n-n by XREAL_1:233
            .= 0;
A24:      n+1-'(n+1)= n+1-(n+1) by XREAL_1:233
            .= 0;
A25:      (Coef(n)).(n) = n! /((n! ) * (0! )) by A23,Def6
            .= 1 by Th1,XCMPLX_1:60;
A26:      (Coef(n+1)).(n+1) = (n+1)! /(((n+1)! ) * (0! )) by A24,Def6
            .= 1 by Th1,XCMPLX_1:60;
      ((w (#) Expan(n,z,w))+Shift((z (#) Expan(n,z,w)))).k =
          (w (#) Expan(n,z,w)).(n+1)+(Shift((z (#) Expan(n,z,w)))).(n+1)
          by A21,VALUED_1:1
.=(w * Expan(n,z,w).(n+1)) + (Shift((z (#) Expan(n,z,w)))).(n+1) by VALUED_1:6
            .=( w * 0c) + (Shift((z (#) Expan(n,z,w)))).(n+1) by A22,Def9
            .=((z (#) Expan(n,z,w))).n by Def8
            .=z * (Expan(n,z,w)).n by VALUED_1:6
            .=z * ((Coef(n)).n * (z |^ n) * (w |^ (n-'n))) by Def9
            .= ((Coef(n)).n * ((z GeoSeq).n * z) * (w |^ (n-'n)))
            .= ((Coef(n+1)).(n+1) * (z |^ (n+1)) * (w |^ (n-'n)))
          by A25,A26,COMSEQ_3:def 1
            .= Expan(n+1,z,w).k by A21,A23,A24,Def9;
hence ((w (#) Expan(n,z,w))+Shift((z (#) Expan(n,z,w)))).k=Expan(n+1,z,w) .
          k;
        end;
    now
          assume
A27:      k < n+1;
A28:      now
            assume
A29:        k=0;
A30:        n-'0 =n-0 by XREAL_1:233;
A31:        n+1-'0 =n+1-0 by XREAL_1:233;
A32:        (Coef(n)).0 = n! /((0! ) * (n! )) by A30,Def6
              .= 1 by Th1,XCMPLX_1:60;
A33:        (Coef(n+1)).0 = (n+1)! /((0! ) * ((n+1)! )) by A31,Def6
              .= 1 by Th1,XCMPLX_1:60;
                    ((w (#) Expan(n,z,w))+Shift((z (#) Expan(n,z,w)))).k =
            (w (#) Expan(n,z,w)).0+(Shift((z (#) Expan(n,z,w)))).0
            by A29,VALUED_1:1
              .=(w * Expan(n,z,w).0) + (Shift((z (#) Expan(n,z,w)))).0
            by VALUED_1:6
              .=(w * Expan(n,z,w).0) + 0c by Def8
              .=w * ((Coef n).0 * (z |^ 0) * (w |^ (n-'0))) by Def9
              .= (Coef n).0 * (z |^ 0) * (((w GeoSeq)).n * w) by A30
              .= (Coef(n+1)).0 * (z |^ 0) * (w |^ (n+1-'0))
            by A31,A32,A33,COMSEQ_3:def 1
              .= Expan(n+1,z,w).k by A29,Def9;
hence ((w (#) Expan(n,z,w))+Shift((z (#) Expan(n,z,w)))).k =Expan(n+1,z,w).
            k;
          end;
      now
            assume
A34:        k<>0;
then A35:        0+1 <= k by INT_1:7;
A36:        k+1-1 <= n+1-1 by A27,INT_1:7;
        k < k+1 by XREAL_1:29;
then         k-1 <= k+1-1 by XREAL_1:9;
then         k-1 <= n by A36,XXREAL_0:2;
then A37:        k-'1 <= n by A35,XREAL_1:233;
A38:        ((w (#) Expan(n,z,w))+Shift((z (#) Expan(n,z,w)))).k
            =(w (#) Expan(n,z,w)).k+(Shift((z (#) Expan(n,z,w)))).k
            by VALUED_1:1
              .=(w * Expan(n,z,w).k) + (Shift((z (#) Expan(n,z,w)))).k
            by VALUED_1:6
              .=(w * Expan(n,z,w).k) + (z (#) Expan(n,z,w)).(k-'1)
            by A34,Th4
              .=(w * Expan(n,z,w).k) + (z * Expan(n,z,w).(k-'1))
            by VALUED_1:6
              .=(w * ((Coef n).k * (z |^ k) * (w |^ (n-'k))))
            + (z * Expan(n,z,w).(k-'1)) by A36,Def9
              .=(w * ((Coef n).k * (z |^ k)) * (w |^ (n-'k)))
            + (z * ((Coef n).(k-'1) * (z |^ (k-'1))
            * (w |^ (n-'(k-'1))))) by A37,Def9
              .=((Coef n).k * (w * (z |^ k) * (w |^ (n-'k)) )
            + ((Coef n).(k-'1) * ((z |^ (k-'1))
            * z * (w |^ (n-'(k-'1))))));
A39:        (n-'k)+1=n-k+1 by A36,XREAL_1:233
              .=n+1-k
              .=n+1-'k by A27,XREAL_1:233;
A40:        n-'(k-'1)=n-(k-'1) by A37,XREAL_1:233
              .=n-(k-1) by A35,XREAL_1:233
              .=n+1-k
              .=n+1-'k by A27,XREAL_1:233;
        (k-'1)+1 =k-1+1 by A35,XREAL_1:233
              .=k;
            then A41:        (
w |^ (n-'k)) * w = w |^ (n-'k+1) & (z |^ (k-'1)) * z = z |^ k by COMSEQ_3:def 1
;
A42:        (Coef n).k +(Coef n).(k-'1) =n! /((k!) * ((n-'k)!))
            +(Coef n).(k-'1) by A36,Def6
              .=n! /((k!) * ((n-'k)!))
            +n! /(((k-'1)!) * (((n-'(k-'1)))! )) by A37,Def6;
A43:        ((k!) * ((n-'k)!)) * (n+1-k+0*<i>)
            =(k!) * (((n-'k)!) * (n+1-k+0*<i>));
A44:        (((k-'1)!) * (((n-'(k-'1)))! )) * (k+0*<i>)
            =(k+0*<i>) * ((k-'1)!) * (((n-'(k-'1)))! )
              .=(k!) * ((n+1-'k)!) by A34,A40,Th2;
        (n+1-k+0*<i>) <>0c by A27;
            then A45:        n ! /((k!) * ((n-'k)!)) =(n! * (n+1-k+0*<i>))
            /(((k!) * ((n-'k)!)) * (n+1-k+0*<i>)) by XCMPLX_1:91
              .=(n! * (n+1-k+0*<i>))/((k!) * ((n+1-'k)!))
            by A36,A43,Th2;
        n! /(((k-'1)!) * (((n-'(k-'1)))! ))
            =(n! * (k+0*<i>))/((k!) * ((n+1-'k)! ))
            by A34,A44,XCMPLX_1:91;
then         (Coef n).k +(Coef n).(k-'1)
            =( n! * (n+1-k+k+(0+0)*<i>)) / ((k!) * ((n+1-'k)!))
            by A42,A45
              .=((n+1)! ) / ((k!) * ((n+1-'k)!)) by Th1
              .=(Coef(n+1)).k by A27,Def6;
then         ((w (#) Expan(n,z,w))+Shift((z (#) Expan(n,z,w)))).k
            =(Coef(n+1)).k * (z |^ k) * (w |^ (n+1-'k)) by A38,A39,A40,A41
              .=Expan(n+1,z,w).k by A27,Def9;
hence ((w (#) Expan(n,z,w))+Shift((z (#) Expan(n,z,w)))).k =Expan(n+1,z,w).
            k;
          end;
          hence ((w (#) Expan(n,z,w))+Shift((z (#) Expan(n,z,w)))).k
          =Expan(n+1,z,w).k by A28;
        end;
        hence ((w (#) Expan(n,z,w))+Shift((z (#) Expan(n,z,w)))).k
        =Expan(n+1,z,w).k by A19,A20,XXREAL_0:1;
      end;
      hence ((w (#) Expan(n,z,w))+Shift((z (#) Expan(n,z,w)))).k
      =Expan(n+1,z,w).k by A13;
    end;
then
A46: (w (#) Expan(n,z,w))+Shift((z (#) Expan(n,z,w))) =Expan(n+1,z,w);
    thus (z+w) |^ (n+1) =(Partial_Sums((w (#) Expan(n,z,w))) +
    Partial_Sums(Shift((z (#) Expan(n,z,w))))).(n+1)
    by A7,A8,A9,A10,A12,VALUED_1:1
      .=Partial_Sums(Expan(n+1,z,w)).(n+1) by A46,COMSEQ_3:27;
  end;
 for n holds X[n] from NAT_1:sch 2(A2,A3);
  hence thesis;
end;
