let s be Real_Sequence; :: thesis: ( ( for n being Nat holds s . n >= 1 ) implies for n being Nat holds (Partial_Product s) . n >= 1 )
defpred S₁[ Nat] means (Partial_Product s) . $1 >= 1;
assume A1: for n being Nat holds s . n >= 1 ; :: thesis: for n being Nat holds (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] )
A3: (Partial_Product s) . (n + 1) = ((Partial_Product s) . n) * (s . (n + 1)) by Def1;
assume A4: (Partial_Product s) . n >= 1 ; :: thesis: S₁[n + 1]
then (Partial_Product s) . n <= ((Partial_Product s) . n) * (s . (n + 1)) by A1, XREAL_1:151;
hence S₁[n + 1] by A4, A3, XXREAL_0:2; :: thesis: verum

end;

(Partial_Product s) . 0 = s . 0 by Def1;
then A5: S₁[ 0 ] by A1;
for n being Nat holds S₁[n] from NAT_1:sch 2(A5, A2);
hence for n being Nat holds (Partial_Product s) . n >= 1 ; :: thesis: verum