reconsider k9 = k as Element of NAT by ORDINAL1:def 12;

deffunc H_{1}( Element of NAT ) -> Element of omega = $1 |^ k9;

consider f being sequence of NAT 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 = n |^ k

for n being Nat holds f . n = n |^ k

deffunc H

consider f being sequence of NAT such that

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

take f ; :: thesis: for n being Nat holds f . n = n |^ k

for n being Nat holds f . n = n |^ k

proof

hence
for n being Nat holds f . n = n |^ k
; :: thesis: verum
let n be Nat; :: thesis: f . n = n |^ k

reconsider n9 = n as Element of NAT by ORDINAL1:def 12;

f . n9 = H_{1}(n9)
by A1;

hence f . n = n |^ k ; :: thesis: verum

end;reconsider n9 = n as Element of NAT by ORDINAL1:def 12;

f . n9 = H

hence f . n = n |^ k ; :: thesis: verum