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 Th17:
  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;
A5: n in NAT by ORDINAL1:def 12;
    now
      let k be Element of NAT;
A6:   now
A7:     now
          assume
A8:       k < n+1;
A9:       now
A10:         ( (k! ) * ((n -' k)! ) ) * (n+1 - k) =(k! ) * (((n -' k)! ) *
            (n+1 - k));
A11:        k+1-1 <= n+1-1 by A8,INT_1:7,A5;
            then
A12:        (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
A13:        n! /( (k! ) * ((n-'k)!) ) =(n! * (n+1-k)) /(( (k! ) * ((n-'k)
            ! ) ) * (n+1-k)) by XCMPLX_1:91
              .=(n! * (n+1-k))/((k! ) * ((n+1-'k)! )) by A11,A10,Th13;
            assume
A14:        k<>0;
            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 A11,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;
A19:        (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;
            (((k-'1)! ) * (((n-'(k-'1)))! )) * k =k * ((k-'1)! ) * (((n-'
            (k-'1)))! )
              .=(k! ) * ((n+1-'k)! ) by A14,A19,Th13;
            then
A20:        n! /(((k-'1)! ) * (((n-'(k-'1)))! )) =(n! * k)/((k! ) * ((n+1
            -'k)! )) by A14,XCMPLX_1:91;
            (( 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 A14,Th15
              .=(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 A11,Def6
              .=(((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,Def6
              .=((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 LOPBAN_3:38
              .=((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 LOPBAN_3:38
              .=((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 LOPBAN_3:38
              .=((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 LOPBAN_3:38
              .=((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
A21:        ((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 A12,A18,A16,LOPBAN_3:38
              .=(Coef(n)).k * ((z #N k) * (w #N (n+1-'k))) + (Coef(n)).((k
            -' 1)) *((z #N k) *(w #N (n+1-'k))) by LOPBAN_3:38
              .= ((Coef(n)).k +(Coef(n)).(k -' 1) ) * ((z #N k) * (w #N ((n+
            1) -' k) )) by LOPBAN_3:38;
            (Coef(n)).k +(Coef(n)).(k -' 1) =n! /( (k! ) * ((n -' k)! ) )
            +(Coef(n)).(k-'1) by A11,Def3
              .=n! /( (k! ) * ((n -' k)! ) ) +n! /(((k-'1)! ) * (((n-'(k-'1)
            ))! )) by A17,Def3;
            then
            (Coef(n)).k +(Coef(n)).(k-'1) =((n! * (n+1-k))+(n! * k)) / ((
            k! ) * ((n+1-'k)! )) by A13,A20,XCMPLX_1:62
              .=( n! * (n+1-k+k)) / ((k! ) * ((n+1-'k)! ))
              .=((n+1)! ) / ((k! ) * ((n+1-'k)! )) by NEWTON:15
              .=(Coef(n+1)).k by A8,Def3;
            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 A21,LOPBAN_3:38
              .=Expan(n+1,z,w).k by A8,Def6;
            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)! /((0! ) * ((n+1)! )) by Def3
              .= 1 by NEWTON:12,XCMPLX_1:60;
A24:        n-'0 =n-0 by XREAL_1:233;
            then
A25:        (Coef(n)).0 = n! /((0! ) * (n! )) by Def3
              .= 1 by NEWTON:12,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 Def5
              .=( Expan(n,z,w).0 *w) by LOPBAN_3:38
              .= ((Coef(n)).0 * (z #N 0) * (w #N (n-' 0)))*w by Def6
              .= (Coef(n)).0 *(z #N 0) *((w #N (n-' 0))*w) by LOPBAN_3:38
              .= (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,Def6;
            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)! ) * (0! )) by Def3
            .= 1 by NEWTON:12,XCMPLX_1:60;
A30:      n-'n= n-n by XREAL_1:233
            .= 0;
          then
A31:      (Coef(n)).(n) = n! /((n! ) * (0! )) by Def3
            .= 1 by NEWTON:12,XCMPLX_1:60;
A32:      n < n+1 by XREAL_1:29;
          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 A32,Def6
            .=( 0.X ) + (Shift(( Expan(n,z,w)*z))).(n+1) by LOPBAN_3:38
            .= (Shift((Expan(n,z,w)*z ))).(n+1) by LOPBAN_3:38
            .= (( Expan(n,z,w)*z)).(n) by Def5
            .= (Expan(n,z,w)).n *z by LOPBAN_3:def 6
            .=((Coef(n)).n * (z #N n) * (w #N (n-' n))) *z by Def6
            .=(Coef(n)).n * (z #N n) * ((w #N (n-' n)) *z) by LOPBAN_3:38
            .=(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 LOPBAN_3:38
            .=(Coef(n)).n *( (z #N n) * z)*(w #N (n-' n)) by LOPBAN_3:38
            .= ((Coef(n+1)).(n+1) * (z #N (n+1)) * (w #N (n-' n))) by A31,A29
,Lm1
            .= Expan(n+1,z,w).k by A33,A30,A28,Def6;
          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 A36,Th15
          .=(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,Def6
          .= 0.X+ ((Expan(n,z,w).(k -' 1))*z) by LOPBAN_3:38
          .=0.X + ( 0.X*z) by A35,A37,Def6
          .=0.X + 0.X by LOPBAN_3:38
          .=0.X by LOPBAN_3:38;
        hence (( Expan(n,z,w)*w)+Shift((Expan(n,z,w)*z))).k =Expan(n+1,z,w).k
        by A34,Def6;
      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,Def6
      .=Partial_Sums((Expan(n,z,w)*w)).n +0.X by LOPBAN_3:38
      .=Partial_Sums((Expan(n,z,w)*w)).n by LOPBAN_3:38;
    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,Def6
      .=Partial_Sums((Expan(n,z,w)*z)).n +0.X by LOPBAN_3:38
      .=Partial_Sums((Expan(n,z,w)*z)).n by LOPBAN_3:38;
    0 +n < n+1 by XREAL_1:29;
    then
A42: Expan(n,z,w).(n+1)=0.X by Def6;
    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 Th16;
    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,LOPBAN_3:38
      .= Partial_Sums(Shift((Expan(n,z,w)*z))).(n+1) by LOPBAN_3:38;
    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 LOPBAN_3:38
        .=( 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 * (z+w) by LOPBAN_3:def 9
      .=(Partial_Sums(Expan(n,z,w))*(z+w)) .n by A3,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,LOPBAN_3:15
      .= 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,LOPBAN_3:15;
  end;
A45: 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 A45,Def6
    .= 1/(1 * 1) * z #N 0 * w #N 0 by A45,Def3,NEWTON:12
    .= (z GeoSeq).0 * w #N 0 by RLVECT_1:def 8
    .= 1.X * (w GeoSeq).0 by LOPBAN_3:def 9
    .= 1.X * 1.X by LOPBAN_3:def 9
    .= 1.X by LOPBAN_3:38;
  then
A46: X[0] by LOPBAN_3:def 9;
  for n holds X[n] from NAT_1:sch 2(A46,A2);
  hence thesis;
end;
