
theorem Th2:
  for n being Ordinal, L being add-associative right_complementable
right_zeroed well-unital distributive non empty doubleLoopStr, p being Series
  of n, L holds 0_(n,L) *' p = 0_(n,L)
proof
  let n be Ordinal, L be add-associative right_complementable right_zeroed
  well-unital distributive non empty doubleLoopStr, p be Series of n, L;
  set Z = 0_(n,L);
  now
    let b be Element of Bags n;
    consider s being FinSequence of L such that
A1: (Z*'p).b = Sum s and
    len s = len decomp b and
A2: for k being Element of NAT st k in dom s ex b1, b2 being bag of n
    st (decomp b)/.k = <*b1, b2*> & s/.k = Z.b1*p.b2 by POLYNOM1:def 10;
    now
      let k be Nat;
      assume k in dom s;
      then consider b1, b2 being bag of n such that
      (decomp b)/.k = <*b1, b2*> and
A3:   s/.k = Z.b1*p.b2 by A2;
      thus s/.k = 0.L*p.b2 by A3,POLYNOM1:22
        .= 0.L;
    end;
    then Sum s = 0.L by MATRLIN:11;
    hence (Z*'p).b = Z.b by A1,POLYNOM1:22;
  end;
  hence thesis by FUNCT_2:63;
end;
