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

theorem
  for x0 be Element of INT.Ring, x be Element of F_Real,
  F be FinSequence of the carrier of Polynom-Ring INT.Ring st x = x0 holds
  (Eval(~^(Sum F))).x = Sum eval(F,x0)
  proof
    set PRI = Polynom-Ring INT.Ring;
    set PRR = Polynom-Ring F_Real;
    let x0 be Element of INT.Ring, x be Element of F_Real,
        F be FinSequence of the carrier of Polynom-Ring INT.Ring;
    assume
A1: x = x0;
    reconsider erFx = eval(^F,x) as FinSequence of F_Real;
    reconsider eIFx = eval(F,x0) as FinSequence of INT.Ring;
A2: erFx = eIFx by A1,Th28;
    (Eval(~^(Sum F))).x
      = eval(~^(Sum F),x) by POLYNOM5:def 13
     .= eval(~(Sum ^F),x) by Th29
     .= Sum eval(^F,x) by Th26
     .= In(Sum eIFx,F_Real) by A2,LIOUVIL2:5,ALGNUM_1:10;
    hence thesis;
  end;
