let s be Real_Sequence; :: thesis: ( ( for n being Nat holds
( s . n > 0 & s . n < 1 ) ) implies for n being Nat holds (Partial_Product s) . n >= ((Partial_Sums s) . n) - n )
defpred S₁[ Nat] means (Partial_Product s) . $1 >= ((Partial_Sums s) . $1) - $1;
assume A1: for n being Nat holds
( s . n > 0 & s . n < 1 ) ; :: thesis: for n being Nat holds (Partial_Product s) . n >= ((Partial_Sums s) . n) - n
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_Product s) . n >= ((Partial_Sums s) . n) - n ; :: thesis: S₁[n + 1]
A4: s . (n + 1) < 1 by A1;
((Partial_Sums s) . n) - (n + 1) < 0 by A1, Th50, XREAL_1:49;
then (s . (n + 1)) * ((((Partial_Sums s) . n) - n) - 1) > 1 * ((((Partial_Sums s) . n) - n) - 1) by A4, XREAL_1:69;
then A5: ((((s . (n + 1)) * ((Partial_Sums s) . n)) - ((s . (n + 1)) * n)) - (s . (n + 1))) + (s . (n + 1)) > ((((Partial_Sums s) . n) - n) - 1) + (s . (n + 1)) by XREAL_1:8;
s . (n + 1) > 0 by A1;
then ((Partial_Product s) . n) * (s . (n + 1)) >= (((Partial_Sums s) . n) - n) * (s . (n + 1)) by A3, XREAL_1:64;
then ((Partial_Product s) . n) * (s . (n + 1)) > (((Partial_Sums s) . n) + (s . (n + 1))) - (n + 1) by A5, XXREAL_0:2;
then (Partial_Product s) . (n + 1) > (((Partial_Sums s) . n) + (s . (n + 1))) - (n + 1) by SERIES_3:def 1;
hence S₁[n + 1] by SERIES_1:def 1; :: thesis: verum

end;

((Partial_Sums s) . 0) - 0 = s . 0 by SERIES_1:def 1;
then A6: S₁[ 0 ] by SERIES_3:def 1;
for n being Nat holds S₁[n] from NAT_1:sch 2(A6, A2);
hence for n being Nat holds (Partial_Product s) . n >= ((Partial_Sums s) . n) - n ; :: thesis: verum