
theorem Th22:
  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
  for E being Enumeration of Roots(<%a,b%>*'p)
  for P being Permutation of dom E
  holds (BRoots(<%a,b%>*'p)(++)E)*P = BRoots(<%a,b%>*'p)(++)(E*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;
    set q = <%a,b%>;
    set B = BRoots(q*'p);
    let E be Enumeration of Roots(q*'p);
    let P be Permutation of dom E;
    len E = len(B(++)E) by Def1;
    then
A1: dom E = dom(B(++)E) by FINSEQ_3:29;
    then reconsider P1 = P as Permutation of dom(B(++)E);
    (B(++)E)*P1 = B(++)(E*P)
    proof
A2:   len(E*P) = len(B(++)(E*P)) by Def1;
A3:   rng P = dom E by FUNCT_2:def 3;
      then
A4:   dom((B(++)E)*P1) = dom(P1) by A1,RELAT_1:27;
A5:   dom(P1) = dom(E*P) by A3,RELAT_1:27;
      hence
A6:   len((B(++)E)*P1) = len(B(++)(E*P)) by A2,A4,FINSEQ_3:29;
      let n be Nat such that
A7:   1 <= n and
A8:   n <= len((B(++)E)*P1);
A9:   n in dom((B(++)E)*P1) by A7,A8,FINSEQ_3:25;
A10:  (B*E).(P1.n) = (B*E*P1).n by A4,A7,A8,FINSEQ_3:25,FUNCT_1:13
      .= (B*(E*P)).n by RELAT_1:36;
A11:  P1.n in rng P1 by A4,A9,FUNCT_1:def 3;
      then
A12:  1 <= P1.n & P1.n <= len(B(++)E) by FINSEQ_3:25;
A13:  E/.(P1.n) = E.(P1.n) by A3,A11,PARTFUN1:def 6
      .= (E*P1).n by A4,A7,A8,FINSEQ_3:25,FUNCT_1:13
      .= (E*P)/.n by A4,A5,A7,A8,FINSEQ_3:25,PARTFUN1:def 6;
      thus ((B(++)E)*P1).n = (B(++)E).(P1.n) by A4,A7,A8,FINSEQ_3:25,FUNCT_1:13
      .= (B*E).(P1.n) * (E/.(P1.n)) by A12,Def1
      .= (B(++)(E*P)).n by A6,A7,A8,A10,A13,Def1;
    end;
    hence thesis;
  end;
