
theorem
  for L being algebraic-closed domRing, p being non-zero Polynomial of L
  for r being FinSequence of L st r is one-to-one & len r = len p-'1 &
  Roots p = rng r holds Sum r = SumRoots p
  proof
    let L be algebraic-closed domRing, p be non-zero Polynomial of L;
    let r be FinSequence of L such that
A1: r is one-to-one and
A2: len r = len p-'1 and
A3: Roots p = rng r;
    set B = BRoots p, s = support B;
    set L1 = (len r) |-> 1;
    consider f being FinSequence of NAT such that
A4: degree B = Sum f & f = B * canFS s by UPROOTS:def 4;
A5: degree B = len p-'1 by UPROOTS:59;
A6: card dom r = card rng r & dom r = Seg len r by A1,CARD_1:70,FINSEQ_1:def 3;
A7: s = Roots p by UPROOTS:def 9;
A8: s c= dom B & rng canFS s = s by PRE_POLY:37,FUNCT_2:def 3;
    then
A9: dom f = dom canFS s by A4,RELAT_1:27;
    then
A10: len f = len canFS s = card s by FINSEQ_3:29,FINSEQ_1:93;
    then
A11: len f = len r by A3,A6,UPROOTS:def 9;
A12: f is len r -element by A10,A6,A7,A3,CARD_1:def 7;
    reconsider E = canFS s as FinSequence of L by A8,FINSEQ_1:def 4;
A13: dom f = dom r by A10,A6,A7,A3,FINSEQ_3:29;
A14: for j being Nat st j in Seg len r holds f.j >= L1.j
    proof
      let j be Nat such that
A15:  j in Seg len r;
A16:  (canFS s).j in s by A13,A9,A6,A8,A15,FUNCT_1:def 3;
      then reconsider c = E.j as Element of L;
      c is_a_root_of p by A16,A7,POLYNOM5:def 10;
      then multiplicity(p,c) >= 1 by UPROOTS:52;
      then B.c >= 1 by UPROOTS:def 9;
      then f.j >= 1 by A13,A6,A4,A15,FUNCT_1:12;
      hence thesis by A15,FINSEQ_2:57;
    end;
A17: Sum L1 = 1*len r by RVSUM_1:80;
A18: len (B(++)E) = len E by Def1;
    for j being Nat st 1 <= j <= len E holds (B(++)E).j = E.j
    proof
      let j be Nat such that
A19:  1 <=j <= len E;
A20:  j in Seg len r by A19,A11,A10;
      then f.j >= L1.j & f.j <= L1.j & L1.j = 1
      by A2,A14,A12,A17,A4,A5,RVSUM_1:83,FINSEQ_2:57;
      then
A21:  f.j = 1 by XXREAL_0:1;
A22:  E/.j = E.j by A20,A13,A9,A6,PARTFUN1:def 6;
      (B(++)E).j = (f.j) * (E/.j) by A4,A19,A18,Def1;
      hence thesis by BINOM:13,A21,A22;
    end;
    then
A23: B(++)E = E by A18,FINSEQ_1:14;
    E,r are_fiberwise_equipotent by A1,A3,A8,UPROOTS:def 9,RFINSEQ:26;
    then ex P be Permutation of dom E st E = r*P by CLASSES1:80,A13,A9;
    hence thesis by A23,RLVECT_2:7,A13,A9,A7;
  end;
