deffunc H₁( Element of NAT ) -> Element of ExtREAL = f . ($1,0);
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 . ($2,($3 + 1)));
consider IT being Function of [:NAT,NAT:],ExtREAL such that
A2: for a being Element of NAT holds
( IT . (a,0) = f0 . a & ( for n being Nat holds IT . (a,(n + 1)) = H₂(IT . (a,n),a,n) ) ) from DBLSEQ_2:sch 1();
take IT ; :: thesis: for n, m being Nat holds
( IT . (n,0) = f . (n,0) & IT . (n,(m + 1)) = (IT . (n,m)) + (f . (n,(m + 1))) )

hereby :: thesis: verum

let n, m be Nat; :: thesis: ( IT . (n,0) = f . (n,0) & IT . (n,(m + 1)) = (IT . (n,m)) + (f . (n,(m + 1))) )
A3: ( n in NAT & m in NAT ) by ORDINAL1:def 12;
then IT . (n,0) = f0 . n by A2;
hence ( IT . (n,0) = f . (n,0) & IT . (n,(m + 1)) = (IT . (n,m)) + (f . (n,(m + 1))) ) by A1, A2, A3; :: thesis: verum

end;