for x being object holds -infty < (Partial_Sums_in_cod2 f) . x

proof

let x be object ; :: thesis: -infty < (Partial_Sums_in_cod2 f) . x

per cases ( x in dom (Partial_Sums_in_cod2 f) or not x in dom (Partial_Sums_in_cod2 f) ) ;

suppose x in dom (Partial_Sums_in_cod2 f) ; :: thesis: -infty < (Partial_Sums_in_cod2 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_cod2 f) . (n,f) > -infty ;
(Partial_Sums_in_cod2 f) . (n,0) = f . (n,0) by DefCSM;
then A2: S₁[ 0 ] by MESFUNC5:def 5;
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 . (n,(k + 1)) > -infty by MESFUNC5:def 5;
(Partial_Sums_in_cod2 f) . (n,(k + 1)) = ((Partial_Sums_in_cod2 f) . (n,k)) + (f . (n,(k + 1))) by DefCSM;
hence S₁[k + 1] by A4, A5, XXREAL_3:17, XXREAL_0:6; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(A2, A3);
then S₁[m] ;
hence -infty < (Partial_Sums_in_cod2 f) . x by A1; :: thesis: verum

end;

suppose not x in dom (Partial_Sums_in_cod2 f) ; :: thesis: -infty < (Partial_Sums_in_cod2 f) . x

hence -infty < (Partial_Sums_in_cod2 f) . x by FUNCT_1:def 2; :: thesis: verum

end;

end;

end;

hence Partial_Sums_in_cod2 f is without-infty by MESFUNC5:def 5; :: thesis: verum