deffunc H_{1}( Nat) -> Element of NAT = In (($1 + k),NAT);

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

let n be Nat; :: thesis: f . n = n + k

n in NAT by ORDINAL1:def 12;

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

hence f . n = n + k ; :: thesis: verum

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

let n be Nat; :: thesis: f . n = n + k

n in NAT by ORDINAL1:def 12;

then f . n = H

hence f . n = n + k ; :: thesis: verum