let f be nonnegative Function of [:NAT,NAT:],ExtREAL; :: thesis: for n, m being Nat st ( for i being Nat st i <= m holds
f . (n,i) is Real ) holds
(Partial_Sums_in_cod2 f) . (n,m) < +infty
let n, m be Nat; :: thesis: ( ( for i being Nat st i <= m holds
f . (n,i) is Real ) implies (Partial_Sums_in_cod2 f) . (n,m) < +infty )
assume A2: for i being Nat st i <= m holds
f . (n,i) is Real ; :: thesis: (Partial_Sums_in_cod2 f) . (n,m) < +infty
defpred S₁[ Nat] means ( $1 <= m implies (Partial_Sums_in_cod2 f) . (n,$1) < +infty );
(Partial_Sums_in_cod2 f) . (n,0) = f . (n,0) by DefCSM;
then (Partial_Sums_in_cod2 f) . (n,0) is Real by A2;
then A4: S₁[ 0 ] by XXREAL_0:9, XREAL_0:def 1;
A5: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A6: S₁[k] ; :: thesis: S₁[k + 1]

now :: thesis: ( k + 1 <= m implies (Partial_Sums_in_cod2 f) . (n,(k + 1)) < +infty )

assume A7: k + 1 <= m ; :: thesis: (Partial_Sums_in_cod2 f) . (n,(k + 1)) < +infty
then A8: ( f . (n,(k + 1)) is Real & f . (n,(k + 1)) >= 0 ) by A2, SUPINF_2:51;
(Partial_Sums_in_cod2 f) . (n,(k + 1)) = ((Partial_Sums_in_cod2 f) . (n,k)) + (f . (n,(k + 1))) by DefCSM;
hence (Partial_Sums_in_cod2 f) . (n,(k + 1)) < +infty by A6, A7, A8, NAT_1:13, XXREAL_3:16, XXREAL_0:4; :: thesis: verum

end;

hence S₁[k + 1] ; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(A4, A5);
hence (Partial_Sums_in_cod2 f) . (n,m) < +infty ; :: thesis: verum