
theorem
  for n being Ordinal, L being add-associative right_complementable
  right_zeroed right_unital distributive non empty doubleLoopStr, p being
  Series of n, L holds p*'0_(n,L) = 0_(n,L)
proof
  let n be Ordinal, L be add-associative right_complementable right_zeroed
  right_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 the carrier of L such that
A1: (p*'Z).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 = p.b1*Z.b2 by Def10;
    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 = p.b1*Z.b2 by A2;
      thus s/.k = p.b1*0.L by A3,Th22
        .= 0.L;
    end;
    then Sum s = 0.L by MATRLIN:11;
    hence (p*'Z).b = Z.b by A1;
  end;
  hence thesis;
end;
