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 Th16:
  for z,w st z,w are_commutative holds (z+w) #N n = Partial_Sums(
  Expan(n,z,w)).n
proof
  let z,w such that
A1: z,w are_commutative;
  defpred X[Nat] means
(z+w) #N $1 = Partial_Sums(Expan($1,z,w)).$1;
A2: for n st X[n] holds X[n+1]
  proof
    let n such that
A3: (z+w) #N n = Partial_Sums(Expan(n,z,w)).n;
A4: n < n+1 by XREAL_1:29;
    now
      let k be Element of NAT;
A5:  n in NAT by ORDINAL1:def 12;
A6:   now
A7:     now
          assume
A8:       k < n+1;
A9:       now
A10:         k+1-1 <= n+1-1 by A8,A5,INT_1:7;
            then
A11:        (n-'k)+1=n-k+1 by XREAL_1:233
              .=n+1-k
              .=(n+1-'k) by A8,XREAL_1:233;
            (n+1-k) <> 0 by A8;
            then
A12:        n! /( (k!) * ((n-'k)!) ) =(n! * (n+1-k))/(((k!) * ((n-'k
            )!)) * (n+1-k+0*<i>)) by XCMPLX_1:91
              .=(n! * (n+1-k))/((k!) * (((n-'k)!) * (n+1-k+0*<i>)))
              .=(n! * (n+1-k))/((k!) * ((n+1-'k)!)) by A10,SIN_COS:2;
            assume
A13:        k<>0;
            then
A14:        0<k;
            then
A15:        0+1 <= k by INT_1:7;
            then
A16:        (k-'1)+1 =k-1+1 by XREAL_1:233
              .=k;
            k < k+1 by XREAL_1:29;
            then k-1 <= k+1-1 by XREAL_1:9;
            then k-1 <= n by A10,XXREAL_0:2;
            then
A17:        k-'1 <= n by A15,XREAL_1:233;
            then
A18:        (n-'(k-'1)) =n-(k-'1) by XREAL_1:233
              .=n-(k-1) by A15,XREAL_1:233
              .=n+1-k
              .=n+1-'k by A8,XREAL_1:233;
            (((k-'1)!) * (((n-'(k-'1)))!)) * k = k * ((k-'1)!) * (((n
            -'(k-'1)))!)
              .=(k+0*<i>) * ((k-'1)!) * (((n-'(k-'1)))!)
              .=(k! ) * ((n+1-'k)! ) by A14,A18,SIN_COS:2;
            then
A19:        n! / (((k-'1)!) * (((n-'(k-'1)))!)) =(n! * k)/((k! ) * (
            (n+1-'k)! )) by A13,XCMPLX_1:91;
            (Coef(n)).k +(Coef(n)).(k -' 1) =n! /( (k!) * ((n-'k)!) )
            +(Coef(n)).(k-'1) by A10,SIN_COS:def 6
              .=n! /( (k!) * ((n-'k)!) ) +n! /(((k-'1)!)*(((n-'(k-'1)))
            !)) by A17,SIN_COS:def 6;
            then
A20:        (Coef(n)).k +(Coef(n)).(k-'1) =((n! * (n+1-k))+(n! * k)) /
            ((k!) * ((n+1-'k)!)) by A12,A19,XCMPLX_1:62
              .= ( n! * (n+1-k+k) ) / ((k!) * ((n+1-'k)!))
              .= ( n! * (n+1-k+k+(0+0)*<i>) ) / ((k!) * ((n+1-'k)!))
              .=((n+1)! ) / ((k! ) * ((n+1-'k)! )) by SIN_COS:1
              .=(Coef(n+1)).k by A8,SIN_COS:def 6;
A21:        (n-'(k-'1))=n-(k-'1) by A17,XREAL_1:233
              .=n-(k-1) by A15,XREAL_1:233
              .=n+1-k
              .=(n+1-'k) by A8,XREAL_1:233;
            (( Expan(n,z,w)*w )+Shift(( Expan(n,z,w)*z))).k =( Expan(n,z,
            w)*w).k+(Shift((Expan(n,z,w)*z))).k by NORMSP_1:def 2
              .=(Expan(n,z,w).k)*w + (Shift(( Expan(n,z,w)*z))).k by
LOPBAN_3:def 6
              .=(Expan(n,z,w).k)*w + (Expan(n,z,w)*z).( k-'1) by A13,
LOPBAN_4:15
              .=(Expan(n,z,w).k)*w + ( Expan(n,z,w).(k-'1))*z by LOPBAN_3:def 6
              .=( (Coef(n)).k * (z #N k) * (w #N (n-' k)) ) *w + ( Expan(n,z
            ,w).(k-'1))*z by A10,Def2
              .=(((Coef(n)).k * (z #N k) * (w #N (n-' k)))) *w + ( ((Coef(n)
            ).(k -' 1) * (z #N (k-'1)) * (w #N (n-' (k -' 1)))))*z by A17,Def2
              .=((Coef(n)).k * ((z #N k)) * ((w #N (n-' k))*w)) + ( ((Coef(n
)).(k -' 1)) * (z #N (k -' 1)) * (w #N (n-' (k-'1)))) *z by GROUP_1:def 3
              .=((Coef(n)).k * ((z #N k)) * ((w #N ((n-' k)+1)))) + ( ((Coef
            (n)).(k -' 1)) * (z #N (k -' 1)) * (w #N (n-' (k-'1)))) *z by Lm1
              .=((Coef(n)).k * ((z #N k)) * ((w #N ((n-' k)+1)))) + ((Coef(n
)).(k -' 1)) * (z #N (k -' 1)) *((w #N (n-' (k-'1))) *z) by GROUP_1:def 3
              .=((Coef(n)).k * ((z #N k)) * ((w #N ((n-' k)+1)))) + ((Coef(n
            )).(k -' 1)) * (z #N (k -' 1)) *(z*(w #N (n-' (k-'1)))) by A1,Lm2
              .=((Coef(n)).k * ((z #N k)) * ((w #N ((n-' k)+1)))) + ((Coef(n
)).(k -' 1)) * (z #N (k -' 1)) *z*(w #N (n-' (k-'1))) by GROUP_1:def 3
              .=((Coef(n)).k * ((z #N k)) * ((w #N ((n-' k)+1)))) + ((Coef(n
            )).(k -'1))*((z #N (k -'1))*z)*(w #N (n-'(k-'1))) by CLOPBAN3:38
              .=((Coef(n)).k * ((z #N k)) * ((w #N ((n-' k)+1)))) + ((Coef(n
            )).(k -'1))*(z #N ((k -'1)+1))*(w #N (n-'(k-'1))) by Lm1;
            then ((Expan(n,z,w)*w)+Shift((Expan(n,z,w)*z))) .k =(Coef(n)).k *
((z #N k) * (w #N (n+1-'k))) + (Coef(n)).((k -' 1)) *(z #N k) *(w #N (n+1-'k))
            by A11,A21,A16,CLOPBAN3:38
              .=(Coef(n)).k * ((z #N k) * (w #N (n+1-'k))) + (Coef(n)).((k
            -' 1)) *((z #N k) *(w #N (n+1-'k))) by CLOPBAN3:38
              .= ((Coef(n)).k +(Coef(n)).(k -' 1) ) * ((z #N k) * (w #N ((n+
            1) -' k) )) by CLOPBAN3:38;
            then
            ((Expan(n,z,w)*w)+Shift((Expan(n,z,w)*z))).k =(Coef(n+1)).k *
            (z #N k) * (w #N (n+1-'k) ) by A20,CLOPBAN3:38
              .=Expan(n+1,z,w).k by A8,Def2;
            hence
            (( Expan(n,z,w)*w)+Shift((Expan(n,z,w)*z))).k = Expan(n+1,z,w
            ) . k;
          end;
          now
A22:        n+1-'0 =n+1-0 by XREAL_1:233;
            then
A23:        (Coef(n+1)).0 = (n+1)! /(1 * ((n+1)!)) by SIN_COS:1,def 6
              .= 1 by XCMPLX_1:60;
A24:        n-'0 =n-0 by XREAL_1:233;
            then
A25:        (Coef(n)).0 = n! /(1 * (n!)) by SIN_COS:1,def 6
              .= 1 by XCMPLX_1:60;
            assume
A26:        k=0;
            then
            (( Expan(n,z,w)*w)+Shift(( Expan(n,z,w)*z))).k = ( Expan(n,z,
            w)*w).0+(Shift((Expan(n,z,w)*z))).0 by NORMSP_1:def 2
              .=(Expan(n,z,w).0 *w) + (Shift(( Expan(n,z,w)*z))).0 by
LOPBAN_3:def 6
              .=( Expan(n,z,w).0 *w ) + 0.X by LOPBAN_4:def 5
              .=( Expan(n,z,w).0 *w) by RLVECT_1:def 4
              .= ((Coef(n)).0 * (z #N 0) * (w #N (n-'0)))*w by Def2
              .= (Coef(n)).0 *(z #N 0) *((w #N (n-' 0))*w) by GROUP_1:def 3
              .= (Coef(n+1)).0 *(z #N 0)* (w #N ((n+1)-'0)) by A24,A22,A25,A23
,Lm1
              .= Expan(n+1,z,w).k by A26,Def2;
            hence
            ((Expan(n,z,w)*w)+Shift(( Expan(n,z,w)*z))).k = Expan(n+1,z,w
            ) . k;
          end;
          hence ((Expan(n,z,w)*w)+Shift((Expan(n,z,w)*z))).k =Expan(n+1,z,w).k
          by A9;
        end;
A27:    now
A28:      n+1-'(n+1)= n+1-(n+1) by XREAL_1:233
            .= 0;
          then
A29:      (Coef(n+1)).(n+1) = (n+1)! /(((n+1)!) * 1) by SIN_COS:1,def 6
            .= 1 by XCMPLX_1:60;
A30:      n < n+1 by XREAL_1:29;
A31:      n-'n= n-n by XREAL_1:233
            .= 0;
          then
A32:      (Coef(n)).n = n! /((n!) * 1) by SIN_COS:1,def 6
            .= 1 by XCMPLX_1:60;
          assume
A33:      k=n+1;
          then
          (( Expan(n,z,w)*w)+Shift((Expan(n,z,w)*z))).k = (Expan(n,z,w)*w
          ).(n+1)+(Shift((Expan(n,z,w)*z ))).(n+1) by NORMSP_1:def 2
            .= (Expan(n,z,w).(n+1) *w) + (Shift((Expan(n,z,w)*z))).(n+1) by
LOPBAN_3:def 6
            .= (0.X*w) + (Shift(( Expan(n,z,w)*z))).(n+1) by A30,Def2
            .= (0.X) + (Shift(( Expan(n,z,w)*z))).(n+1) by CLOPBAN3:38
            .= (Shift((Expan(n,z,w)*z ))).(n+1) by RLVECT_1:def 4
            .= (( Expan(n,z,w)*z)).(n) by LOPBAN_4:def 5
            .= (Expan(n,z,w)).n *z by LOPBAN_3:def 6
            .=((Coef(n)).n * (z #N n) * (w #N (n-' n))) *z by Def2
            .=(Coef(n)).n * (z #N n) * ((w #N (n-' n)) *z) by GROUP_1:def 3
            .=(Coef(n)).n * (z #N n) * ( z*(w #N (n-' n))) by A1,Lm2
            .=(Coef(n)).n * (z #N n) * z*(w #N (n-' n)) by GROUP_1:def 3
            .=(Coef(n)).n *( (z #N n) * z)*(w #N (n-' n)) by CLOPBAN3:38
            .= ((Coef(n+1)).(n+1) * (z #N (n+1)) * (w #N (n-' n))) by A32,A29
,Lm1
            .= Expan(n+1,z,w).k by A33,A31,A28,Def2;
          hence
          ((Expan(n,z,w)*w) +Shift((Expan(n,z,w)*z))).k= Expan(n+1,z,w).k;
        end;
        assume k <= (n+1);
        hence
        ((Expan(n,z,w)*w)+Shift((Expan(n,z,w)*z))).k =Expan(n+1,z,w).k by A27
,A7,XXREAL_0:1;
      end;
      now
        assume
A34:    n+1 < k;
        then
A35:    n+1-1 < k -1 by XREAL_1:9;
        then
A36:    n+0 < k-1+1 by XREAL_1:8;
    0+1 <= n+1 by XREAL_1:6;
        then
A37:    k-1 =k-'1 by A34,XREAL_1:233,XXREAL_0:2;
        (( Expan(n,z,w)*w)+Shift(( Expan(n,z,w)*z))).k =(Expan(n,z,w)*w).
        k + (Shift((Expan(n,z,w)*z))).k by NORMSP_1:def 2
          .=(Expan(n,z,w).k*w) + (Shift(( Expan(n,z,w)*z))).k by LOPBAN_3:def 6
          .=(Expan(n,z,w).k*w) + ((Expan(n,z,w)*z)).(k -' 1) by A34,LOPBAN_4:15
          .=(Expan(n,z,w).k*w) + ((Expan(n,z,w).(k -' 1)*z)) by LOPBAN_3:def 6
          .=( 0.X*w )+ ((Expan(n,z,w).(k -' 1))*z) by A36,Def2
          .= 0.X+ ((Expan(n,z,w).(k -' 1))*z) by CLOPBAN3:38
          .=0.X + ( 0.X*z) by A35,A37,Def2
          .=0.X + 0.X by CLOPBAN3:38
          .=0.X by RLVECT_1:def 4;
        hence (( Expan(n,z,w)*w)+Shift((Expan(n,z,w)*z))).k =Expan(n+1,z,w).k
        by A34,Def2;
      end;
      hence ((Expan(n,z,w)*w)+Shift((Expan(n,z,w)*z))).k =Expan(n+1,z,w).k by
A6;
    end;
    then
A38: ( Expan(n,z,w)*w)+Shift(( Expan(n,z,w)*z)) =Expan(n+1,z,w) by FUNCT_2:63;
A39: n < n+1 by XREAL_1:29;
    Partial_Sums((Expan(n,z,w)*w)).(n+1) =Partial_Sums((Expan(n,z,w)*w)).
    n+(Expan(n,z,w)*w).(n+1) by BHSP_4:def 1
      .=Partial_Sums((Expan(n,z,w)*w)).n + Expan(n,z,w).(n+1) *w by
LOPBAN_3:def 6;
    then
A40: Partial_Sums(( Expan(n,z,w)*w)).(n+1) =Partial_Sums((Expan(n,z,w)*w))
    .n +0.X *w by A39,Def2
      .=Partial_Sums((Expan(n,z,w)*w)).n +0.X by CLOPBAN3:38
      .=Partial_Sums((Expan(n,z,w)*w)).n by RLVECT_1:def 4;
    Partial_Sums((Expan(n,z,w)*z)).(n+1) = Partial_Sums((Expan(n,z,w)*z))
    .n+(Expan(n,z,w)*z).(n+1) by BHSP_4:def 1
      .= Partial_Sums((Expan(n,z,w)*z)).n + Expan(n,z,w).(n+1) *z by
LOPBAN_3:def 6;
    then
A41: Partial_Sums(( Expan(n,z,w)*z)).(n+1) =Partial_Sums((Expan(n,z,w)*z))
    .n +0.X *z by A4,Def2
      .=Partial_Sums((Expan(n,z,w)*z)).n +0.X by CLOPBAN3:38
      .=Partial_Sums((Expan(n,z,w)*z)).n by RLVECT_1:def 4;
    0 +n < n+1 by XREAL_1:29;
    then
A42: Expan(n,z,w).(n+1)=0.X by Def2;
    Partial_Sums(( Expan(n,z,w)*z)).(n+1) =Partial_Sums(Shift(( Expan(n,z
    ,w)*z))).(n+1) +( Expan(n,z,w)*z).(n+1) by Th15;
    then
A43: Partial_Sums(( Expan(n,z,w)*z)).(n+1) =Partial_Sums(Shift(( Expan(n,z
    ,w)*z))).(n+1) + Expan(n,z,w).(n+1)*z by LOPBAN_3:def 6
      .=Partial_Sums(Shift(( Expan(n,z,w)*z))).(n+1) +0.X by A42,CLOPBAN3:38
      .= Partial_Sums(Shift((Expan(n,z,w)*z))).(n+1) by RLVECT_1:def 4;
    now
      let k be Element of NAT;
      thus ( Expan(n,z,w)*(z+w) ).k= Expan(n,z,w).k *(z+w) by LOPBAN_3:def 6
        .= Expan(n,z,w).k * z + Expan(n,z,w).k *w
          by VECTSP_1:def 2
        .=( Expan(n,z,w)*z).k+ Expan(n,z,w).k *w by LOPBAN_3:def 6
        .=(Expan(n,z,w)*z).k+(Expan(n,z,w)*w).k by LOPBAN_3:def 6
        .=( (Expan(n,z,w)*z)+(Expan(n,z,w)*w)).k by NORMSP_1:def 2;
    end;
    then
A44: Expan(n,z,w)*(z+w) =( Expan(n,z,w)*z)+(Expan(n,z,w)*w) by FUNCT_2:63;
    (z+w) #N (n+1) =((z+w) GeoSeq ).(n+1) by CLOPBAN3:def 8
      .=((z+w) GeoSeq ).n * (z+w) by CLOPBAN3:def 7
      .=(Partial_Sums(Expan(n,z,w)).n) * (z+w) by A3,CLOPBAN3:def 8
      .=(Partial_Sums(Expan(n,z,w))*(z+w)) .n by LOPBAN_3:def 6
      .=(Partial_Sums(Expan(n,z,w)*(z+w))).n by Th9;
    then (z+w) #N (n+1) =( Partial_Sums( (Expan(n,z,w)*z)) +Partial_Sums(( (
    Expan(n,z,w)*w)))).n by A44,CLOPBAN3:15
      .= Partial_Sums((Expan(n,z,w)*z)).n +Partial_Sums((Expan(n,z,w)*w)).n
    by NORMSP_1:def 2;
    hence
    (z+w) #N (n+1) =(Partial_Sums(( Expan(n,z,w)*w)) + Partial_Sums(Shift
    (( Expan(n,z,w)*z)))).(n+1) by A41,A40,A43,NORMSP_1:def 2
      .=Partial_Sums(Expan(n+1,z,w)).(n+1) by A38,CLOPBAN3:15;
  end;
A45: (z+w) #N 0 =((z+w) GeoSeq).0 by CLOPBAN3:def 8
    .=1.X by CLOPBAN3:def 7;
A46: 0-'0=0-0 by XREAL_0:def 2
    .=0;
  Partial_Sums(Expan(0,z,w)).0 = Expan(0,z,w).0 by BHSP_4:def 1
    .= (Coef(0)).0 * (z #N 0) * (w #N 0) by A46,Def2
    .= 1r/(1r * 1r) * z #N 0 * w #N 0 by A46,COMPLEX1:def 4,SIN_COS:1,def 6
    .= z #N 0 * w #N 0 by CLVECT_1:def 5,COMPLEX1:def 4
    .= (z GeoSeq).0 * w #N 0 by CLOPBAN3:def 8
    .= (z GeoSeq).0 * (w GeoSeq).0 by CLOPBAN3:def 8
    .= 1.X * (w GeoSeq).0 by CLOPBAN3:def 7
    .= 1.X * 1.X by CLOPBAN3:def 7
    .= 1.X by VECTSP_1:def 4;
  then
A47: X[0] by A45;
  for n holds X[n] from NAT_1:sch 2(A47,A2);
  hence thesis;
end;
