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 Th5:
    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
    holds E_eval(Sum F,x) = Sum E_eval(F,x)
    proof
      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;
      per cases;
        suppose
A1:       len F = 0; then
A2:       F = <*>(the carrier of Polynom-Ring(n,L));
A3:       0.Polynom-Ring(n,L) = 0_(n,L) by POLYNOM1:def 11;
A4:       E_eval(Sum F,x) = E_eval(0.Polynom-Ring(n,L),x) by A2,RLVECT_1:43
          .= eval(0_(n,L),x) by A3,Def1 .= 0.L by POLYNOM2:20;
          Seg len F = dom F by FINSEQ_1:def 3
          .= dom E_eval(F,x) by Def2
          .= Seg(len E_eval(F,x)) by FINSEQ_1:def 3; then
          E_eval(F,x) = <*>(the carrier of L) by A1;
          hence thesis by A4,RLVECT_1:43;
        end;
        suppose
A5:       len F <> 0;
          for k be non zero Nat holds
          len F = k implies E_eval(Sum F,x) = Sum E_eval(F,x)
          proof
            let k be non zero Nat;
            defpred P[Nat] means
            for F be FinSequence of the carrier of Polynom-Ring(n,L) st
            len F = $1 holds E_eval(Sum F,x) = Sum E_eval(F,x);
A6:         P[1]
            proof
              for F be FinSequence of the carrier of Polynom-Ring(n,L) st
              len F = 1 holds E_eval(Sum F,x) = Sum E_eval(F,x)
              proof
                let F be FinSequence of the carrier of Polynom-Ring(n,L);
                assume
A7:             len F = 1; then
                dom F = Seg 1 by FINSEQ_1:def 3; then
A9:             1 in dom F; then
                F.1 in rng F by FUNCT_1:3; then
                reconsider o = F.1 as Element of Polynom-Ring(n,L);
                F = <*o*> by A7,FINSEQ_1:40; then
A11:            Sum F = F.1 by BINOM:3 .= F/.1 by A9,PARTFUN1:def 6;
A12:            dom E_eval(F,x) = dom F by Def2 .= Seg 1
                  by A7,FINSEQ_1:def 3;
                set o1 = E_eval(F,x).1;
                set o = E_eval(F,x)/.1;
A13:            1 in dom E_eval(F,x) by A12;
A14:            dom E_eval(F,x) = dom F by Def2;
                E_eval(F,x) = <* o1 *> by A12,FINSEQ_1:def 8 .= <* o *>
                  by A13,PARTFUN1:def 6; then
                Sum E_eval(F,x) = E_eval(F,x).1 by BINOM:3
                  .= E_eval(Sum F,x) by A11,A13,A14,Def2;
                hence thesis;
              end;
              hence thesis;
            end;
A15:        for k be non zero Nat holds P[k] implies P[k+1]
            proof
              let k be non zero Nat;
              assume
A16:          P[k];
              for F be FinSequence of the carrier of Polynom-Ring(n,L)
              st len F = k+1 holds E_eval(Sum F,x) = Sum E_eval(F,x)
              proof
                let F be FinSequence of the carrier of Polynom-Ring(n,L);
                assume
A17:            len F = k+1; then
                consider G be FinSequence of Polynom-Ring(n,L),
                d be Element of Polynom-Ring(n,L) such that
A18:            F = G^<*d*> by FINSEQ_2:19;
                (F|k)^<* F/.len F *> = G^<*d*> by A18,A17,FINSEQ_5:21; then
A20:            G = (F|k) & d = F/.len F by FINSEQ_2:17;
A21:            k+ 1 = len G + 1 by FINSEQ_2:16,A18,A17;
                Sum F = Sum G + d by A18,FVSUM_1:71; then
                E_eval(Sum F,x) = E_eval(Sum G,x) + E_eval(d,x) by Th2
                .= Sum(E_eval(F|k,x)) + E_eval(F/.len F,x) by A21,A16,A20
                .= Sum(E_eval(F|k,x)^<* E_eval(F/.(len F),x) *>) by FVSUM_1:71
                .= Sum(E_eval(F,x)) by A17,Th4;
                hence thesis;
              end;
              hence P[k + 1];
            end;
            for k being non zero Nat holds P[k] from NAT_1:sch 10(A6,A15);
            hence thesis;
          end;
          hence thesis by A5;
        end;
      end;
