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