 reserve n,k for Nat;
 reserve L for comRing;
 reserve R for domRing;
 reserve x0 for positive Real;

theorem
  for F be non empty FinSequence of the carrier of Polynom-Ring INT.Ring,
      x be Element of F_Real
    holds eval(~(^(Product F)),x) = Product eval(^F,x)
    proof
       let F be non empty FinSequence of the carrier of Polynom-Ring INT.Ring,
           x be Element of F_Real;
A1:    Seg len F = dom F by FINSEQ_1:def 3 .= dom ^F by Def7
       .= Seg len ^F by FINSEQ_1:def 3;
reconsider F2 = ^F as non empty FinSequence of the carrier of
Polynom-Ring F_Real by A1;
       eval(~(^(Product F)),x) = eval(~(Product F2),x) by Lm38
       .= Product eval(F2,x) by Th39;
       hence thesis;
    end;
