reserve A for non degenerated comRing;
reserve R for non degenerated domRing;
reserve n for non empty Ordinal;
reserve o,o1,o2 for object;
reserve X,Y for Subset of Funcs(n,[#]R);
reserve S,T for Subset of Polynom-Ring(n,R);
reserve F,G for FinSequence of the carrier of Polynom-Ring(n,R);
reserve x for Function of n,R;

theorem Th4:
    for N0 be Nat
    for n be Ordinal,
    L being right_zeroed add-associative
    right_complementable Abelian well-unital distributive
    non trivial commutative associative non empty doubleLoopStr,
    F be FinSequence of the carrier of Polynom-Ring(n,L),
    x be Function of n,L
    st len F = N0+1 holds
    E_eval(F,x) = (E_eval(F|N0,x))^<*E_eval(F/.(len F),x)*>
    proof
      let N0 be Nat;
      let n be Ordinal,
      L be right_zeroed add-associative
      right_complementable Abelian well-unital distributive
      non trivial commutative associative non empty doubleLoopStr,
      F be FinSequence of the carrier of Polynom-Ring(n,L),
      x be Function of n,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 E_eval(F,x) by Def2
      .= Seg len E_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 E_eval(F|N0,x)) = dom E_eval(F|N0,x) by FINSEQ_1:def 3
      .= dom (F|N0) by Def2 .= Seg N0 by A4,FINSEQ_1:def 3; then
A6:   len E_eval(F|N0,x) = N0 by FINSEQ_1:6;
      len <*E_eval(F/.(N0+1),x)*> = 1 by FINSEQ_1:39; then
A8:   len (E_eval(F|N0,x)^<*E_eval(F/.(N0+1),x)*>) = N0+1 by A6,FINSEQ_1:22;
      for k be Nat st 1 <= k & k <= len E_eval(F,x) holds
      E_eval(F,x).k = (E_eval(F|N0,x)^<*E_eval(F/.(len F),x)*>).k
      proof
        let k be Nat;
        assume 1 <= k & k <= len E_eval(F,x); then
        k in Seg (N0+1) by A3; then
A11:    k in Seg N0 \/ {N0+1} by FINSEQ_1:9;
A13:    Seg N0 c= Seg(N0 + 1) by FINSEQ_3:18;
        per cases by A11,XBOOLE_0:def 3;
          suppose
A12:        k in Seg N0; then
A14:        k in dom (F|N0) by A4,FINSEQ_1:def 3; then
A15:        k in dom E_eval(F|N0,x) by Def2;
A16:        k in dom (F|Seg N0) by A12,A4,FINSEQ_1:def 3;
A17:        (F|N0)/.k = (F|Seg N0).k by A14,PARTFUN1:def 6
            .= F.k by A16,FUNCT_1:47 .= F/.k by A2,A12,A13,PARTFUN1:def 6;
            ((E_eval(F|N0,x))^<* E_eval(F/.(N0+1),x) *>).k
            = (E_eval(F|N0,x)).k by A15,FINSEQ_1:def 7
            .= E_eval(F/.k,x) by A17,A14,Def2
            .= E_eval(F,x).k by Def2,A2,A12,A13;
            hence thesis by A1;
          end;
          suppose k in {N0+1}; then
A19:         k = N0+1 by TARSKI:def 1;
             N0+1 = len E_eval(F|N0,x) + 1 by A5,FINSEQ_1:6;
             hence thesis by A1,Def2,A2,FINSEQ_1:4,A19;
           end;
         end;
         hence thesis by A1,A3,A8,FINSEQ_1:6;
       end;
