deffunc H1( Element of NAT ) -> Element of ExtREAL = f . (0,$1);
consider f0 being Function of NAT,ExtREAL such that
A1:
for n being Element of NAT holds f0 . n = H1(n)
from FUNCT_2:sch 4();
deffunc H2( Element of ExtREAL , Nat, Nat) -> Element of ExtREAL = $1 + (f . (($3 + 1),$2));
consider IT being Function of [:NAT,NAT:],ExtREAL such that
A2:
for a being Element of NAT holds
( IT . (0,a) = f0 . a & ( for n being Nat holds IT . ((n + 1),a) = H2(IT . (n,a),a,n) ) )
from DBLSEQ_2:sch 3();
take
IT
; for n, m being Nat holds
( IT . (0,m) = f . (0,m) & IT . ((n + 1),m) = (IT . (n,m)) + (f . ((n + 1),m)) )
hereby verum
let n,
m be
Nat;
( IT . (0,m) = f . (0,m) & IT . ((n + 1),m) = (IT . (n,m)) + (f . ((n + 1),m)) )A3:
(
n in NAT &
m in NAT )
by ORDINAL1:def 12;
then
IT . (
0,
m)
= f0 . m
by A2;
hence
(
IT . (
0,
m)
= f . (
0,
m) &
IT . (
(n + 1),
m)
= (IT . (n,m)) + (f . ((n + 1),m)) )
by A1, A2, A3;
verum
end;