reserve i,j,n,n1,n2,m,k,l,u for Nat,
        r,r1,r2 for Real,
        x,y for Integer,
        a,b for non trivial Nat,
        F for XFinSequence,
        cF,cF1,cF2 for complex-valued XFinSequence,
        c,c1,c2 for Complex;

theorem
  for X being natural-valued positive-yielding XFinSequence st
    i in dom X holds X.i <= Product X
proof
  defpred P[Nat] means for X be natural-valued positive-yielding XFinSequence,
    i be Nat st len X=$1 & i in dom X holds X.i <= Product X;
A1: P[0] by XBOOLE_0:def 1;
A2: P[n] implies P[n+1]
    proof
      set n1=n+1;
      assume
A3:     P[n];
      let X be positive-yielding natural-valued XFinSequence, i be Nat;
      assume
A4:     len X=n1 & i in dom X;
      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;
A6:   n < n1 by NAT_1:13;
      then
A7:     n in dom X & len (X|n) = n by A4,AFINSQ_1:54,66;
      then Product (X|n) > 0 & X.n in rng X by FUNCT_1:def 3;
      then
A8:     X.n >0 & Product (X|n) >=1 by PARTFUN3:def 1,NAT_1:14;
      then
A9:     X.n >=1 by NAT_1:14;
      i < len X by A4,AFINSQ_1:66;
      then i <=n by A4,NAT_1:13;
      then per cases by XXREAL_0:1;
      suppose i =n;
        then Product X >= X.i*1 by A5,A8,XREAL_1:66;
        hence thesis;
      end;
      suppose
A10:      i<n;
        then
A11:      i in dom (X|n) by A7,AFINSQ_1:66;
        Product (X|n) >= (X|n).i by A3,A7,AFINSQ_1:66,A10;
        then Product X >= ((X|n).i)*1 & (X|n).i = X.i
          by A6,A4,AFINSQ_1:53,A11,A7,A5,A9,XREAL_1:66;
        hence thesis;
      end;
    end;
A12: P[n] from NAT_1:sch 2(A1,A2);
    let X be natural-valued positive-yielding XFinSequence;
    thus thesis by A12;
end;
