let s, s1 be Real_Sequence; :: thesis: ( ( for n being Element of NAT holds
( s . n > 0 & s1 . n = 1 / (s . n) ) ) implies for n being Element of NAT holds (Partial_Sums s1) . n > 0 )
assume A1: for n being Element of NAT holds
( s . n > 0 & s1 . n = 1 / (s . n) ) ; :: thesis: for n being Element of NAT holds (Partial_Sums s1) . n > 0
defpred S₁[ Element of NAT ] means (Partial_Sums s1) . $1 > 0 ;
A2: s . 0 > 0 by A1;
A3: (Partial_Sums s1) . 0 = s1 . 0 by SERIES_1:def 1;
s1 . 0 = 1 / (s . 0 ) by A1;
then A4: S₁[ 0 ] by A2, A3;
A5: 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 A6: (Partial_Sums s1) . n > 0 ; :: thesis: S₁[n + 1]
A7: (Partial_Sums s1) . (n + 1) = ((Partial_Sums s1) . n) + (s1 . (n + 1)) by SERIES_1:def 1;
A8: s1 . (n + 1) = 1 / (s . (n + 1)) by A1;
s . (n + 1) > 0 by A1;
then s1 . (n + 1) > 0 by A8;
hence S₁[n + 1] by A6, A7; :: thesis: verum

end;

for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A4, A5);
hence for n being Element of NAT holds (Partial_Sums s1) . n > 0 ; :: thesis: verum