deffunc H_{1}( Nat) -> Element of ExtREAL = (H . $1) . x;

consider f being sequence of ExtREAL such that

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

take f ; :: thesis: for n being Nat holds f . n = (H . n) . x

let n be Nat; :: thesis: f . n = (H . n) . x

n in NAT by ORDINAL1:def 12;

hence f . n = (H . n) . x by A1; :: thesis: verum

consider f being sequence of ExtREAL such that

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

take f ; :: thesis: for n being Nat holds f . n = (H . n) . x

let n be Nat; :: thesis: f . n = (H . n) . x

n in NAT by ORDINAL1:def 12;

hence f . n = (H . n) . x by A1; :: thesis: verum