deffunc H_{1}( Nat) -> Element of the carrier of X = (1r / ($1 !)) * (z #N $1);

consider s being sequence of X such that

A1: for n being Element of NAT holds s . n = H_{1}(n)
from FUNCT_2:sch 4();

take s ; :: thesis: for n being Nat holds s . n = (1r / (n !)) * (z #N n)

let n be Nat; :: thesis: s . n = (1r / (n !)) * (z #N n)

n in NAT by ORDINAL1:def 12;

hence s . n = (1r / (n !)) * (z #N n) by A1; :: thesis: verum

consider s being sequence of X such that

A1: for n being Element of NAT holds s . n = H

take s ; :: thesis: for n being Nat holds s . n = (1r / (n !)) * (z #N n)

let n be Nat; :: thesis: s . n = (1r / (n !)) * (z #N n)

n in NAT by ORDINAL1:def 12;

hence s . n = (1r / (n !)) * (z #N n) by A1; :: thesis: verum