for x being object holds +infty > (Partial_Sums_in_cod1 f) . x

proof

let x be object ; :: thesis: +infty > (Partial_Sums_in_cod1 f) . x

per cases ( x in dom (Partial_Sums_in_cod1 f) or not x in dom (Partial_Sums_in_cod1 f) ) ;

suppose x in dom (Partial_Sums_in_cod1 f) ; :: thesis: +infty > (Partial_Sums_in_cod1 f) . x

then consider n, m being object such that
A1: ( n in NAT & m in NAT & x = [n,m] ) by ZFMISC_1:def 2;
reconsider n = n, m = m as Element of NAT by A1;
defpred S₁[ Nat] means (Partial_Sums_in_cod1 f) . (f,m) < +infty ;
(Partial_Sums_in_cod1 f) . (0,m) = f . (0,m) by DefRSM;
then A2: S₁[ 0 ] by MESFUNC5:def 6;
A3: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A4: S₁[k] ; :: thesis: S₁[k + 1]
A5: f . ((k + 1),m) < +infty by MESFUNC5:def 6;
(Partial_Sums_in_cod1 f) . ((k + 1),m) = ((Partial_Sums_in_cod1 f) . (k,m)) + (f . ((k + 1),m)) by DefRSM;
hence S₁[k + 1] by A4, A5, XXREAL_3:16, XXREAL_0:4; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(A2, A3);
then S₁[n] ;
hence +infty > (Partial_Sums_in_cod1 f) . x by A1; :: thesis: verum

end;

suppose not x in dom (Partial_Sums_in_cod1 f) ; :: thesis: +infty > (Partial_Sums_in_cod1 f) . x

hence +infty > (Partial_Sums_in_cod1 f) . x by FUNCT_1:def 2; :: thesis: verum

end;

end;

end;

hence Partial_Sums_in_cod1 f is without+infty by MESFUNC5:def 6; :: thesis: verum