let n be Element of NAT ; :: thesis: for s being Real_Sequence st ( for n being Element of NAT holds s . n > 0 ) holds
(Partial_Product s) . n > 0
let s be Real_Sequence; :: thesis: ( ( for n being Element of NAT holds s . n > 0 ) implies (Partial_Product s) . n > 0 )
defpred S₁[ Element of NAT ] means (Partial_Product s) . $1 > 0 ;
assume A1: for n being Element of NAT holds s . n > 0 ; :: thesis: (Partial_Product s) . n > 0
A2: 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 A3: (Partial_Product s) . n > 0 ; :: 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 A3; :: thesis: verum

end;

s . 0 > 0 by A1;
then A4: S₁[ 0 ] by Def1;
for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A4, A2);
hence (Partial_Product s) . n > 0 ; :: thesis: verum