
theorem Th24:
  for L being Field
  for p being non-zero Polynomial of L
  for a being Element of L
  for b being non zero Element of L st not -a/b in Roots(p)
  for E being Enumeration of Roots(<%a,b%>*'p) st E = (canFS Roots(p))^<*-a/b*>
  holds
  Sum(BRoots(<%a,b%>*'p)(++)E) =
  Sum(BRoots(<%a,b%>*'p)(++)canFS Roots(<%a,b%>*'p))
  proof
    let L be Field;
    let p be non-zero Polynomial of L;
    let a be Element of L;
    let b be non zero Element of L such that
A1: not -a/b in Roots(p);
    set q = <%a,b%>;
    set B = BRoots(q*'p);
    set C = canFS Roots(p);
    set D = canFS Roots(q*'p);
    let E be Enumeration of Roots(q*'p);
    assume E = C^<*-a/b*>;
    then reconsider P = D"*E as Permutation of dom E by A1,Th23;
    len(B(++)E) = len E by Def1;
    then
A2: dom(B(++)E) = dom E by FINSEQ_3:29;
    D"" = D by FUNCT_1:43;
    then
A3: P" = E"*D by FUNCT_1:44;
A4: E*E"*D = D
    proof
A5:   rng D = Roots(q*'p) by FUNCT_2:def 3;
A6:   rng E = Roots(q*'p) by RLAFFIN3:def 1;
      dom(E*E") = rng E by FUNCT_1:37;
      hence
A7:   dom(E*E"*D) = dom D by A5,A6,RELAT_1:27;
      let x be object such that
A8:   x in dom(E*E"*D);
      D.x in rng E by A5,A6,A7,A8,FUNCT_1:def 3;
      hence D.x = (E*E").(D.x) by FUNCT_1:35
      .= (E*E"*D).x by A8,FUNCT_1:12;
    end;
    (B(++)E)*P" = B(++)(E*P") by Th22;
    hence Sum(B(++)E) = Sum(B(++)(E*P")) by A2,RLVECT_2:7
    .= Sum(B(++)D) by A3,A4,RELAT_1:36;
  end;
