
theorem Th38:
  for L being add-associative right_zeroed right_complementable
  distributive well-unital commutative associative non degenerated non empty
doubleLoopStr, r being Element of L, p, q being Polynomial of L st p = <%r, 1.
  L%>*'q & p.(len p -'1) = 1.L holds q.(len q -'1) = 1.L
proof
  let L be add-associative right_zeroed right_complementable distributive
well-unital commutative associative non degenerated non empty doubleLoopStr,
  x be Element of L, p, q be Polynomial of L such that
A1: p = <%x, 1.L%>*'q and
A2: p.(len p -'1) = 1.L;
  set lp1 = len p -'1;
A3: now
    assume q = 0_. L;
    then p = 0_. L by A1,POLYNOM3:34;
    hence contradiction by A2,FUNCOP_1:7;
  end;
  then q is non-zero;
  then len p = len q + 1 by A1,Th35;
  then
A4: lp1 = len q by NAT_D:34;
  then consider lp2 being Nat such that
A5: lp1 = lp2+1 by A3,NAT_1:6,POLYNOM4:5;
  reconsider lp2 as Element of NAT by ORDINAL1:def 12;
A6: q.lp1 = 0.L by A4,ALGSEQ_1:8;
  (<%x, 1.L%>*'q).lp1 = x*q.(lp1)+(1.L)*q.lp2 by A5,Th34
    .= 0.L +(1.L)*q.lp2 by A6
    .= (1.L)*q.lp2 by RLVECT_1:4
    .= q.lp2;
  hence thesis by A1,A2,A4,A5,NAT_D:34;
end;
