reserve k,m,n for Nat;
reserve R for commutative Ring,
        p,q for Polynomial of R,
        z0,z1 for Element of R;

theorem Th7:
  for L being right_zeroed add-associative right_complementable
    right-distributive non empty doubleLoopStr
  for z being Element of L, p being Polynomial of L
    holds (p*' <%z%>).n = (p.n) * z
proof
  let L be right_zeroed add-associative right_complementable
    right-distributive non empty doubleLoopStr,
    z be Element of L, p be Polynomial of L;
  set Z = <%z%>;
  n in NAT by ORDINAL1:def 12;
  then consider r be FinSequence of the carrier of L such that
  A1: len r = n+1 and
  A2: (p*' <%z%>).n = Sum r and
  A3: for k be Element of NAT st k in dom r holds r.k = p.(k-'1) * Z.(n+1-'k)
    by POLYNOM3:def 9;
  set l=len r;
  A4: 1<=l by A1,NAT_1:11;
  then A5: l in dom r & l-'1 =l-1 & n+1-'l =0
    by A1,FINSEQ_3:25,XREAL_1:233,232;
  then A6:r.l = p.n * Z.(n+1-'l) & Z.(n+1-'l) = z by A1,A3,POLYNOM5:32;
  for k be Element of NAT st k in dom r & k<>l holds r/.k = 0.L
  proof
    let k be Element of NAT such that A7: k in dom r & k<>l;
    A8:r/.k=r.k by PARTFUN1:def 6,A7;
    k <= l by A7,FINSEQ_3:25;
    then l-'k = l-k by XREAL_1:233;
    then l-'k <>0 by A7;
    then Z.(l-'k) =0.L by NAT_1:14,POLYNOM5:32;
    then p.(k-'1) * Z.(l-'k) =0.L;
    hence thesis by A1,A8,A7,A3;
  end;
  then Sum r = r/.l by A4,FINSEQ_3:25,POLYNOM2:3;
  hence thesis by A6,A2,A5,PARTFUN1:def 6;
end;
