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

theorem Th29:
  for F be FinSequence of the carrier of Polynom-Ring INT.Ring holds
    Sum ^F = ^(Sum F)
   proof
     let F be FinSequence of the carrier of Polynom-Ring INT.Ring;
     set L = INT.Ring;
     set PRI = Polynom-Ring INT.Ring;
A1:  Seg len ^F = dom ^F by FINSEQ_1:def 3
     .= dom F by Def7 .= Seg len F by FINSEQ_1:def 3;
     per cases;
       suppose
A2:      len F = 0; then
         F = <*>(the carrier of Polynom-Ring L); then
A4:      Sum F = 0.Polynom-Ring L by RLVECT_1:43;
         ^F = <*>(the carrier of Polynom-Ring F_Real) by A2,A1; then
         Sum ^F = 0.Polynom-Ring F_Real by RLVECT_1:43;
         hence thesis by A4,FIELD_4:11;
       end;
       suppose
A6:      len F <> 0;
         for k be non zero Nat holds
         len F = k implies Sum ^F = ^(Sum F)
         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 Sum ^F = ^(Sum F);
A7:        P[1]
           proof
             for F be FinSequence of the carrier of Polynom-Ring L st
             len F = 1 holds Sum ^F = ^(Sum F)
             proof
               let F be FinSequence of the carrier of Polynom-Ring L;
               assume
A8:            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 F1 = F.1 as Element of Polynom-Ring L;
A10:           Seg len ^F = dom ^F by FINSEQ_1:def 3
               .= dom F by Def7 .= Seg len F by FINSEQ_1:def 3;
A11:           F = <*F1*> by A8,FINSEQ_1:40;
               F/.1 = F1 by A9,PARTFUN1:def 6; then
A13:           (^F).1 = ^F1 by A9,Def7;
reconsider RF1 = ^F1 as Element of Polynom-Ring F_Real;
reconsider RF = ^F as FinSequence of the carrier of Polynom-Ring F_Real;
A14:           RF = <* ^F1 *> by A13,A8,A10,FINSEQ_1:6,FINSEQ_1:40;
               Sum RF = ^F1 by BINOM:3,A14;
               hence thesis by BINOM:3,A11;
             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 L
             st len F = k+1 holds Sum ^F = ^(Sum F)
             proof
               let F be FinSequence of the carrier of Polynom-Ring L;
               assume
A17:           len F = k+1; then
               consider G be FinSequence of Polynom-Ring L,
               d be Element of Polynom-Ring L such that
A18:           F = G^<*d*> by FINSEQ_2:19;
A19:           Seg len F = dom F by FINSEQ_1:def 3
               .= dom ^F by Def7 .= Seg len ^F by FINSEQ_1:def 3;
               dom F = Seg (k+1) by A17,FINSEQ_1:def 3; then
A20:           dom ^F = Seg(k+1) by Def7;
               (F|k)^<* F/.len F *> = G^<*d*> by A18,A17,FINSEQ_5:21; then
A21:           G = (F|k) & d = F/.len F by FINSEQ_2:17;
A22:           k+ 1 = len G + 1 by FINSEQ_2:16,A18,A17;
reconsider RF = ^F as FinSequence of the carrier of Polynom-Ring F_Real;
               len F in Seg len F by A17,FINSEQ_1:3; then
A23:           len F in dom F by FINSEQ_1:def 3;
reconsider Fl = F/.len F as Element of the carrier of PRI;
A24:           len RF = len F by A19,FINSEQ_1:6;
               len RF in dom F by A19,FINSEQ_1:6,A23; then
               len RF in dom RF by Def7; then
A25:           RF/.(len RF) = RF.(len RF) by PARTFUN1:def 6
               .= (^F).(len F) by A19,FINSEQ_1:6
               .= ^Fl by A23,Def7;
               Sum F = Sum G + d by A18,FVSUM_1:71; then
               ^Sum F = ^Sum G + ^d by Th27
               .= Sum(^(F|k)) + ^(F/.len F) by A22,A16,A21
               .= Sum((^F)|k) + ^Fl by A17,Lm16
               .= Sum(((^F)|k)^<* ^Fl *>) by FVSUM_1:71
               .= Sum(RF|(k+1)) by A24,A17,FINSEQ_5:82,A25
               .= Sum ^F by A20;
               hence thesis;
             end;
             hence P[k + 1];
           end;
           for k being non zero Nat holds P[k] from NAT_1:sch 10(A7,A15);
           hence thesis;
         end;
         hence thesis by A6;
       end;
     end;
