let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n > 0 ) implies for n being Nat holds (Partial_Sums s) . n > 0 )
defpred S1[ Nat] means (Partial_Sums s) . $1 > 0 ;
assume A1: for n being Nat holds s . n > 0 ; :: thesis: for n being Nat holds (Partial_Sums s) . n > 0
A2: for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be Nat; :: thesis: ( S1[n] implies S1[n + 1] )
A3: (Partial_Sums s) . (n + 1) = ((Partial_Sums s) . n) + (s . (n + 1)) by SERIES_1:def 1;
assume (Partial_Sums s) . n > 0 ; :: thesis: S1[n + 1]
hence S1[n + 1] by A1, A3, XREAL_1:34; :: thesis: verum
end;
(Partial_Sums s) . 0 = s . 0 by SERIES_1:def 1;
then A4: S1[ 0 ] by A1;
for n being Nat holds S1[n] from NAT_1:sch 2(A4, A2);
 hence for n being Nat holds (Partial_Sums s) . n > 0 ; :: thesis: verum