let X be Banach_Algebra; :: thesis: for z, w being Element of X
for n being Element of NAT holds Expan_e (n,z,w) = (1 / (n !)) * (Expan (n,z,w))
let z, w be Element of X; :: thesis: for n being Element of NAT holds Expan_e (n,z,w) = (1 / (n !)) * (Expan (n,z,w))
let n be Element of NAT ; :: thesis: Expan_e (n,z,w) = (1 / (n !)) * (Expan (n,z,w))

now

let k be Element of NAT ; :: thesis: (Expan_e (n,z,w)) . k = ((1 / (n !)) * (Expan (n,z,w))) . k

A1: now

n ! <> 0 by NEWTON:23;
then A2: 1 / ((k !) * ((n -' k) !)) = (((n !) * 1) / (n !)) / ((k !) * ((n -' k) !)) by XCMPLX_1:60
.= ((1 / (n !)) * (n !)) / ((k !) * ((n -' k) !)) by XCMPLX_1:75 ;
assume A3: k <= n ; :: thesis: ( (Expan_e (n,z,w)) . k = ((1 / ((k !) * ((n -' k) !))) * (z #N k)) * (w #N (n -' k)) & (Expan_e (n,z,w)) . k = ((1 / (n !)) * (Expan (n,z,w))) . k )
hence (Expan_e (n,z,w)) . k = (((Coef_e n) . k) * (z #N k)) * (w #N (n -' k)) by Def7
.= ((1 / ((k !) * ((n -' k) !))) * (z #N k)) * (w #N (n -' k)) by A3, Def4 ;
:: thesis: (Expan_e (n,z,w)) . k = ((1 / (n !)) * (Expan (n,z,w))) . k
hence (Expan_e (n,z,w)) . k = (((1 / (n !)) * (n !)) / ((k !) * ((n -' k) !))) * ((z #N k) * (w #N (n -' k))) by A2, LOPBAN_3:43
.= ((1 / (n !)) * ((n !) / ((k !) * ((n -' k) !)))) * ((z #N k) * (w #N (n -' k))) by XCMPLX_1:75
.= (1 / (n !)) * (((n !) / ((k !) * ((n -' k) !))) * ((z #N k) * (w #N (n -' k)))) by LOPBAN_3:43
.= (1 / (n !)) * ((((n !) / ((k !) * ((n -' k) !))) * (z #N k)) * (w #N (n -' k))) by LOPBAN_3:43
.= (1 / (n !)) * ((((Coef n) . k) * (z #N k)) * (w #N (n -' k))) by A3, Def3
.= (1 / (n !)) * ((Expan (n,z,w)) . k) by A3, Def6
.= ((1 / (n !)) * (Expan (n,z,w))) . k by NORMSP_1:def 8 ;
:: thesis: verum

end;

now

assume A4: n < k ; :: thesis: (Expan_e (n,z,w)) . k = ((1 / (n !)) * (Expan (n,z,w))) . k
hence (Expan_e (n,z,w)) . k = 0. X by Def7
.= (1 / (n !)) * (0. X) by LOPBAN_3:43
.= (1 / (n !)) * ((Expan (n,z,w)) . k) by A4, Def6
.= ((1 / (n !)) * (Expan (n,z,w))) . k by NORMSP_1:def 8 ;
:: thesis: verum

end;

hence (Expan_e (n,z,w)) . k = ((1 / (n !)) * (Expan (n,z,w))) . k by A1; :: thesis: verum

end;

hence Expan_e (n,z,w) = (1 / (n !)) * (Expan (n,z,w)) by FUNCT_2:113; :: thesis: verum