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 > 0 & (Partial_Product s) . n < 1 ) )
defpred S₁[ Nat] means ( (Partial_Product s) . $1 > 0 & (Partial_Product 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 > 0 & (Partial_Product s) . n < 1 )
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 > 0 & (Partial_Product s) . n < 1 ) ; :: thesis: S₁[n + 1]
( (Partial_Product s) . (n + 1) = ((Partial_Product s) . n) * (s . (n + 1)) & s . (n + 1) > 0 ) by A1, Def1;
hence S₁[n + 1] by A1, A3, XREAL_1:162; :: thesis: verum

end;

(Partial_Product s) . 0 = s . 0 by Def1;
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_Product s) . n > 0 & (Partial_Product s) . n < 1 ) ; :: thesis: verum