let s be Real_Sequence; :: thesis: ( ( for n being Element of NAT holds s . n >= 0 ) implies for n being Element of NAT holds (Partial_Sums s) . n >= 0 )
assume A1: for n being Element of NAT holds s . n >= 0 ; :: thesis: for n being Element of NAT holds (Partial_Sums s) . n >= 0
defpred S₁[ Element of NAT ] means (Partial_Sums s) . $1 >= 0 ;
(Partial_Sums s) . 0 = s . 0 by SERIES_1:def 1;
then A2: S₁[ 0 ] by A1;
A3: for n being Element of NAT st S₁[n] holds
S₁[n + 1]

proof

let n be Element of NAT ; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume A4: (Partial_Sums s) . n >= 0 ; :: thesis: S₁[n + 1]
A5: (Partial_Sums s) . (n + 1) = ((Partial_Sums s) . n) + (s . (n + 1)) by SERIES_1:def 1;
s . (n + 1) >= 0 by A1;
hence S₁[n + 1] by A4, A5; :: thesis: verum

end;

for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A2, A3);
hence for n being Element of NAT holds (Partial_Sums s) . n >= 0 ; :: thesis: verum