let s, s1 be Real_Sequence; :: thesis: ( s = s1 (#) s1 implies for n being Nat holds (Partial_Sums s) . n >= 0 )
defpred S₁[ Nat] means (Partial_Sums s) . $1 >= 0 ;
assume A1: s = s1 (#) s1 ; :: 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] )
assume A3: (Partial_Sums s) . n >= 0 ; :: thesis: S₁[n + 1]
A4: (s1 . (n + 1)) ^2 >= 0 by XREAL_1:63;
(Partial_Sums s) . (n + 1) = ((Partial_Sums s) . n) + (s . (n + 1)) by SERIES_1:def 1
.= ((Partial_Sums s) . n) + ((s1 . (n + 1)) ^2) by A1, SEQ_1:8 ;
hence S₁[n + 1] by A3, A4; :: thesis: verum

end;

(Partial_Sums s) . 0 = s . 0 by SERIES_1:def 1
.= (s1 . 0) ^2 by A1, SEQ_1:8 ;
then A5: S₁[ 0 ] by XREAL_1:63;
for n being Nat holds S₁[n] from NAT_1:sch 2(A5, A2);
hence for n being Nat holds (Partial_Sums s) . n >= 0 ; :: thesis: verum