let f be nonnegative Function of [:NAT,NAT:],ExtREAL; :: thesis: for n, m being Nat holds
( (Partial_Sums_in_cod1 f) . (n,m) >= f . (n,m) & (Partial_Sums_in_cod2 f) . (n,m) >= f . (n,m) )
let n, m be Nat; :: thesis: ( (Partial_Sums_in_cod1 f) . (n,m) >= f . (n,m) & (Partial_Sums_in_cod2 f) . (n,m) >= f . (n,m) )
defpred S₁[ Nat] means ( $1 <= n implies (Partial_Sums_in_cod1 f) . ($1,m) >= f . ($1,m) );
A2: S₁[ 0 ] by DefRSM;
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 S₁[k] ; :: thesis: S₁[k + 1]
assume k + 1 <= n ; :: thesis: (Partial_Sums_in_cod1 f) . ((k + 1),m) >= f . ((k + 1),m)
(Partial_Sums_in_cod1 f) . ((k + 1),m) = ((Partial_Sums_in_cod1 f) . (k,m)) + (f . ((k + 1),m)) by DefRSM;
hence (Partial_Sums_in_cod1 f) . ((k + 1),m) >= f . ((k + 1),m) by SUPINF_2:51, XXREAL_3:39; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(A2, A5);
hence (Partial_Sums_in_cod1 f) . (n,m) >= f . (n,m) ; :: thesis: (Partial_Sums_in_cod2 f) . (n,m) >= f . (n,m)
defpred S₂[ Nat] means ( $1 <= m implies (Partial_Sums_in_cod2 f) . (n,$1) >= f . (n,$1) );
A2: S₂[ 0 ] by DefCSM;
A5: for k being Nat st S₂[k] holds
S₂[k + 1]

proof

let k be Nat; :: thesis: ( S₂[k] implies S₂[k + 1] )
assume S₂[k] ; :: thesis: S₂[k + 1]
assume k + 1 <= m ; :: thesis: (Partial_Sums_in_cod2 f) . (n,(k + 1)) >= f . (n,(k + 1))
(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)) >= f . (n,(k + 1)) by SUPINF_2:51, XXREAL_3:39; :: thesis: verum

end;

for k being Nat holds S₂[k] from NAT_1:sch 2(A2, A5);
hence (Partial_Sums_in_cod2 f) . (n,m) >= f . (n,m) ; :: thesis: verum