
theorem Th49:
  for L be Abelian add-associative right_zeroed
  right_complementable well-unital commutative distributive non empty
  doubleLoopStr for p be Polynomial of L holds Subst(0_.(L),p) = 0_.(L)
proof
  let L be Abelian add-associative right_zeroed right_complementable
  well-unital commutative distributive non empty doubleLoopStr;
  let p be Polynomial of L;
  consider F be FinSequence of the carrier of Polynom-Ring L such that
A1: Subst(0_.(L),p) = Sum F and
  len F = len (0_.(L)) and
A2: for n be Element of NAT st n in dom F holds F.n = (0_.(L)).(n-'1)*(p
  `^(n-'1)) by Def6;
  now
    let n be Element of NAT;
    assume n in dom F;
    hence F.n = (0_.(L)).(n-'1)*(p`^(n-'1)) by A2
      .= 0.L*(p`^(n-'1)) by FUNCOP_1:7
      .= 0_.(L) by Th26
      .= 0.(Polynom-Ring L) by POLYNOM3:def 10;
  end;
  hence Subst(0_.(L),p) = 0.(Polynom-Ring L) by A1,POLYNOM3:1
    .= 0_.(L) by POLYNOM3:def 10;
end;
