
theorem Th12:
  for n being Ordinal, L being add-associative right_zeroed
  right_complementable well-unital associative commutative distributive non
empty doubleLoopStr, p,p9 being Series of n,L, a being Element of L holds a *
  (p *' p9) = p *' (a * p9)
proof
  let n be Ordinal, L be add-associative right_zeroed right_complementable
well-unital commutative associative distributive non empty doubleLoopStr, p,
  p9 be Series of n,L, a be Element of L;
  set app = a * (p *' p9), pap = p *' (a * p9), pp = p *' p9;
A1: now
    let u be object;
    assume u in dom app;
    then reconsider b = u as bag of n;
    consider s being FinSequence of the carrier of L such that
A2: pap.b = Sum s and
A3: len s = len decomp b and
A4: 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*(a*p9).b2 by POLYNOM1:def 10;
    consider t being FinSequence of the carrier of L such that
A5: pp.b = Sum t and
A6: len t = len decomp b and
A7: for k being Element of NAT st k in dom t ex b1,b2 being bag of n
    st (decomp b)/.k = <*b1, b2*> & t/.k = p.b1*(p9).b2 by POLYNOM1:def 10;
A8: dom t = Seg(len s) by A3,A6,FINSEQ_1:def 3
      .= dom s by FINSEQ_1:def 3;
    now
      let i be object;
      assume
A9:   i in dom t;
      then reconsider k = i as Element of NAT;
      consider b1,b2 being bag of n such that
A10:  (decomp b)/.k = <*b1,b2*> and
A11:  t/.k = p.b1*(p9).b2 by A7,A9;
      consider a1,a2 being bag of n such that
A12:  (decomp b)/.k = <*a1,a2*> and
A13:  s/.k = p.a1*(a*p9).a2 by A4,A8,A9;
A14:  b2 = <*a1,a2*>.2 by A10,A12
        .= a2;
      b1 = <*a1,a2*>.1 by A10,A12
        .= a1;
      hence s/.i = p.b1 * (a*(p9).b2) by A13,A14,POLYNOM7:def 9
        .= a*(t/.i) by A11,GROUP_1:def 3;
    end;
    then s = a * t by A8,POLYNOM1:def 1;
    then pap.b = a * Sum t by A2,POLYNOM1:12
      .= app.b by A5,POLYNOM7:def 9;
    hence app.u = pap.u;
  end;
  dom app = Bags n by FUNCT_2:def 1
    .= dom pap by FUNCT_2:def 1;
  hence thesis by A1,FUNCT_1:2;
end;
