defpred S₁[ Nat] means for X being real-valued positive-yielding XFinSequence st len X = X holds
Product X is positive ;
A1: S₁[ 0 ]

proof

let X be real-valued positive-yielding XFinSequence; :: thesis: ( len X = 0 implies Product X is positive )
assume len X = 0 ; :: thesis: Product X is positive
then X = {} ;
hence Product X is positive by Th4; :: thesis: verum

end;

A2: for n being Nat st S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
set n1 = n + 1;
assume A3: S₁[n] ; :: thesis: S₁[n + 1]
let X be real-valued positive-yielding XFinSequence; :: thesis: ( len X = n + 1 implies Product X is positive )
set XX = <%(X . n)%>;
assume A4: len X = n + 1 ; :: thesis: Product X is positive
then X = (X | n) ^ <%(X . n)%> by AFINSQ_1:56;
then A5: Product X = (Product (X | n)) * (Product <%(X . n)%>) by Th7
.= (Product (X | n)) * (X . n) by Th5 ;
n < n + 1 by NAT_1:13;
then ( n in dom X & len (X | n) = n ) by A4, AFINSQ_1:54, AFINSQ_1:66;
then A6: ( Product (X | n) > 0 & X . n in rng X ) by A3, FUNCT_1:def 3;
then X . n > 0 by PARTFUN3:def 1;
hence Product X is positive by A5, A6; :: thesis: verum

end;

for n being Nat holds S₁[n] from NAT_1:sch 2(A1, A2);
then S₁[ len X] ;
hence Product X is positive ; :: thesis: verum