deffunc H₁( Element of NAT ) -> Element of REAL = Rseq . (0,$1);
consider f0 being Function of NAT,REAL such that
a0: for n being Element of NAT holds f0 . n = H₁(n) from FUNCT_2:sch 4();
deffunc H₂( Real, Nat, Nat) -> set = $1 + (Rseq . (($3 + 1),$2));
consider IT being Function of [:NAT,NAT:],REAL such that
a1: for a being Element of NAT holds
( IT . (0,a) = f0 . a & ( for n being natural number holds IT . ((n + 1),a) = H₂(IT . (n,a),a,n) ) ) from DBLSEQ_2:sch 4();
take IT ; :: thesis: for n, m being Nat holds
( IT . (0,m) = Rseq . (0,m) & IT . ((n + 1),m) = (IT . (n,m)) + (Rseq . ((n + 1),m)) )

hereby :: thesis: verum

let n, m be natural number ; :: thesis: ( IT . (0,m) = Rseq . (0,m) & IT . ((n + 1),m) = (IT . (n,m)) + (Rseq . ((n + 1),m)) )
a2: ( n in NAT & m in NAT ) by ORDINAL1:def 12;
then IT . (0,m) = f0 . m by a1;
hence ( IT . (0,m) = Rseq . (0,m) & IT . ((n + 1),m) = (IT . (n,m)) + (Rseq . ((n + 1),m)) ) by a0, a1, a2; :: thesis: verum

end;