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

theorem Th25:
  for N0 be Nat
  for L being comRing,
    F be FinSequence of the carrier of Polynom-Ring L, x be Element of L
    st len F = N0+1 holds
    eval(F,x) = (eval(F|N0,x))^<*eval(~(F/.(len F)),x)*>
    proof
      let N0 be Nat;
      let L be comRing,
      F be FinSequence of the carrier of Polynom-Ring L,
      x be Element of L;
      assume
A1:   len F = N0+1; then
A2:   dom F = Seg(N0+1) by FINSEQ_1:def 3; then
A3:   Seg (N0+1) = dom eval(F,x) by Def8
      .= Seg len eval(F,x) by FINSEQ_1:def 3;
A4:   len(F|N0) = min(N0,len F) by FINSEQ_2:21 .= N0 by A1;
A5:   Seg len eval(F|N0,x) = dom eval(F|N0,x) by FINSEQ_1:def 3
      .= dom (F|N0) by Def8 .= Seg N0 by A4,FINSEQ_1:def 3; then
A6:   len eval(F|N0,x) = N0 by FINSEQ_1:6;
      len <* eval(~(F/.(N0+1)),x)*> = 1 by FINSEQ_1:39; then
A7:   len (eval(F|N0,x)^<*eval(~(F/.(N0+1)),x)*>) = N0+1 by A6,FINSEQ_1:22;
      for k be Nat st 1 <= k & k <= len eval(F,x) holds
      eval(F,x).k = (eval(F|N0,x)^<* eval(~(F/.(len F)),x)*>).k
      proof
        let k be Nat;
        assume 1 <= k & k <= len eval(F,x); then
        k in Seg (N0+1) by A3; then
A8:     k in Seg N0 \/ {N0+1} by FINSEQ_1:9;
A9:     Seg N0 c= Seg(N0 + 1) by FINSEQ_3:18;
        per cases by A8,XBOOLE_0:def 3;
          suppose
A10:        k in Seg N0; then
A11:        k in dom (F|N0) by A4,FINSEQ_1:def 3; then
A12:        k in dom eval(F|N0,x) by Def8;
A13:        k in dom (F|Seg N0) by A10,A4,FINSEQ_1:def 3;
A14:        (F|N0)/.k = (F|Seg N0).k by A11,PARTFUN1:def 6
            .= F.k by A13,FUNCT_1:47 .= F/.k by A2,A10,A9,PARTFUN1:def 6;
            ((eval(F|N0,x))^<* eval(~(F/.(N0+1)),x) *>).k
            = (eval(F|N0,x)).k by A12,FINSEQ_1:def 7
            .= eval(~(F/.k),x) by A14,A11,Def8
            .= eval(F,x).k by Def8,A2,A10,A9;
            hence thesis by A1;
          end;
          suppose k in {N0+1}; then
A15:         k = N0+1 by TARSKI:def 1;
             N0+1 = len eval(F|N0,x) + 1 by A5,FINSEQ_1:6;
             hence thesis by A1,Def8,A2,FINSEQ_1:4,A15;
           end;
         end;
         hence thesis by A1,A3,A7,FINSEQ_1:6;
       end;
