
theorem Th20:
  for L be Abelian add-associative right_zeroed
  right_complementable unital distributive non empty doubleLoopStr for p be
  Polynomial of L for x be Element of L holds eval(-p,x) = -eval(p,x)
proof
  let L be Abelian add-associative right_zeroed right_complementable unital
  distributive non empty doubleLoopStr;
  let p be Polynomial of L;
  let x be Element of L;
  consider F1 be FinSequence of the carrier of L such that
A1: eval(p,x) = Sum F1 and
A2: len F1 = len p and
A3: for n be Element of NAT st n in dom F1 holds F1.n = p.(n-'1) * (
  power L).(x,n-'1) by Def2;
  consider F2 be FinSequence of the carrier of L such that
A4: eval(-p,x) = Sum F2 and
A5: len F2 = len (-p) and
A6: for n be Element of NAT st n in dom F2 holds F2.n=(-p).(n-'1)*(power
  L).(x,n-'1) by Def2;
  len F2 = len F1 by A2,A5,Th8;
  then
A7: dom F2 = dom F1 by FINSEQ_3:29;
  now
    let n be Nat;
    let v be Element of L;
    assume that
A8: n in dom F2 and
A9: v = F1.n;
    thus F2.n = (-p).(n-'1)*(power L).(x,n-'1) by A6,A8
      .= (-p.(n-'1))*(power L).(x,n-'1) by BHSP_1:44
      .= -p.(n-'1)*(power L).(x,n-'1) by VECTSP_1:9
      .= -v by A3,A7,A8,A9;
  end;
  hence thesis by A1,A2,A4,A5,Th8,RLVECT_1:40;
end;
