
theorem Th30:
  for R being Abelian add-associative right_zeroed
  right_complementable associative distributive well-unital commutative non
  trivial non empty doubleLoopStr, n being Nat, p being Element of
  Polynom-Ring (n+1,R) holds upm(n,R).(mpu(n,R).p) = p
proof
  let R being Abelian add-associative right_zeroed right_complementable
  associative distributive well-unital commutative non trivial
doubleLoopStr, n being Nat, p being Element of Polynom-Ring (n+1,R);
  set PNR = Polynom-Ring(n,R);
  reconsider p9=p as Polynomial of (n+1), R by POLYNOM1:def 11;
  reconsider upmmpup = upm(n,R).(mpu(n,R).p) as Polynomial of (n+1), R by
POLYNOM1:def 11;
  reconsider mpup = (mpu(n,R).p) as Polynomial of PNR by POLYNOM3:def 10;
  now
    let b9 be object;
    assume b9 in Bags (n+1);
    then reconsider b=b9 as Element of Bags (n+1);
    reconsider mpupbn = mpup.(b.n) as Polynomial of n, R by POLYNOM1:def 11;
    n < n+1 by NAT_1:13;
    then reconsider bn = (b|n) as bag of n by Th3;
A1: b = bn bag_extend b.n by Def1;
    thus upmmpup.b9 = mpupbn.bn by Def6
      .= p9.b9 by A1,Def7;
  end;
  hence thesis by FUNCT_2:12;
end;
