
theorem Th27:
  for L being Field
  for a being Element of L
  for b being non zero Element of L holds
  SumRoots <%a,b%> = -a/b
  proof
    let L be Field;
    let a be Element of L;
    let b be non zero Element of L;
    set p = <%a,b%>;
    set B = BRoots(p);
A1: Roots p = {-a/b} by Th10;
    reconsider E = canFS Roots p as Enumeration of Roots p by Th2;
    set F = B(++)E;
    consider g being sequence of L such that
A2: Sum(F) = g.(len F) and
A3: g.0 = 0.L and
A4: for j being Nat, v being Element of L st j < len F & v = F.(j+1)
    holds g.(j+1) = g.j + v by RLVECT_1:def 12;
A5: E = <*-a/b*> by A1,FINSEQ_1:94;
A6: len F = len E by Def1;
A7: len E = 1 by A5,FINSEQ_1:39;
A8: 1 in dom E by A7,FINSEQ_3:25;
A9: (B*E).1 = B.(E.1) by A8,FUNCT_1:13
     .= multiplicity(p,-a/b) by A5,UPROOTS:def 9
     .= 1 by Th11;
A10: F.1 = (B*E).1*(E/.1) by A6,A7,Def1
    .= E/.1 by A9,BINOM:13;
    then reconsider v = F.1 as Element of L;
A11: 0 < len F by A7,Def1;
    thus SumRoots p = g.(0+1) by A2,A7,Def1
    .= g.0 + v by A4,A11
    .= -a/b by A5,A3,A10,FINSEQ_4:16;
  end;
