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

theorem Th26:
  for L being comRing,
      F be FinSequence of the carrier of Polynom-Ring L,
      x be Element of L holds
    eval(~Sum F,x) = Sum eval(F,x)
    proof
      let L be comRing,
          F be FinSequence of the carrier of Polynom-Ring L,
          x be Element of L;
      per cases;
        suppose
A1:       len F = 0; then
A2:       F = <*>(the carrier of Polynom-Ring L);
          0.Polynom-Ring L = 0_.L by POLYNOM3:def 10; then
A4:       eval(~Sum F,x) = eval(0_.L,x) by A2,RLVECT_1:43 .= 0.L
          by POLYNOM4:17;
          Seg len F = dom F by FINSEQ_1:def 3
          .= dom eval(F,x) by Def8
          .= Seg(len eval(F,x)) by FINSEQ_1:def 3; then
          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 eval(~Sum F,x) = Sum eval(F,x)
          proof
            let k be non zero Nat;
            defpred P[Nat] means
            for F be FinSequence of the carrier of Polynom-Ring L st
            len F = $1 holds eval(~Sum F,x) = Sum eval(F,x);
A6:         P[1]
            proof
              for F be FinSequence of the carrier of Polynom-Ring L st
              len F = 1 holds eval(~Sum F,x) = Sum eval(F,x)
              proof
                let F be FinSequence of the carrier of Polynom-Ring L;
                assume
A7:             len F = 1; then
                dom F = Seg 1 by FINSEQ_1:def 3; then
A8:             1 in dom F; then
                F.1 in rng F by FUNCT_1:3; then
                reconsider o = F.1 as Element of Polynom-Ring L;
                F = <*o*> by A7,FINSEQ_1:40; then
A9:             Sum F = F.1 by BINOM:3 .= F/.1 by A8,PARTFUN1:def 6;
A10:            dom eval(F,x) = dom F by Def8 .= Seg 1 by A7,FINSEQ_1:def 3;
                set o1 = eval(F,x).1;
                set o = eval(F,x)/.1;
A11:            1 in dom eval(F,x) by A10;
A12:            dom eval(F,x) = dom F by Def8;
                eval(F,x) = <* o1 *> by A10,FINSEQ_1:def 8 .= <* o *>
                  by A11,PARTFUN1:def 6; then
                Sum eval(F,x) = eval(F,x).1 by BINOM:3
                  .= eval(~Sum F,x) by A9,A11,A12,Def8;
                hence thesis;
              end;
              hence thesis;
            end;
A13:        for k be non zero Nat holds P[k] implies P[k+1]
            proof
              let k be non zero Nat;
              assume
A14:          P[k];
              for F be FinSequence of the carrier of Polynom-Ring L
              st len F = k+1 holds eval(~Sum F,x) = Sum eval(F,x)
              proof
                let F be FinSequence of the carrier of Polynom-Ring L;
                assume
A15:            len F = k+1; then
                consider G be FinSequence of Polynom-Ring L,
                d be Element of Polynom-Ring L such that
A16:            F = G^<*d*> by FINSEQ_2:19;
                (F|k)^<* F/.len F *> = G^<*d*> by A16,A15,FINSEQ_5:21; then
A17:            G = (F|k) & d = F/.len F by FINSEQ_2:17;
A18:            k+ 1 = len G + 1 by FINSEQ_2:16,A16,A15;
                Sum F = Sum G + d by A16,FVSUM_1:71; then
                eval(~Sum F,x) = eval(~Sum G,x) + eval(~d,x) by Lm15
                .= Sum(eval(F|k,x)) + eval(~(F/.len F),x) by A18,A14,A17
                .= Sum(eval(F|k,x)^<* eval(~(F/.(len F)),x) *>) by FVSUM_1:71
                .= Sum(eval(F,x)) by A15,Th25;
                hence thesis;
              end;
              hence P[k + 1];
            end;
            for k being non zero Nat holds P[k] from NAT_1:sch 10(A6,A13);
            hence thesis;
          end;
          hence thesis by A5;
        end;
      end;
