
theorem Th11:
  for X being set, L being associative non empty multLoopStr_0,
p being Series of X,L, a,a9 being Element of L holds (a * a9) * p = a * (a9 * p
  )
proof
  let n be set, L be associative non empty multLoopStr_0, p be Series of n,L
  , a,a9 be Element of L;
  set q = (a * a9) * p, r = a * (a9 * p);
A1: now
    let u be object;
    assume u in dom q;
    then reconsider b = u as bag of n;
    q.b = (a * a9) * p.b by POLYNOM7:def 9
      .= a * (a9 * p.b) by GROUP_1:def 3
      .= a * (a9*p).b by POLYNOM7:def 9
      .= r.b by POLYNOM7:def 9;
    hence q.u = r.u;
  end;
  dom q = Bags n by FUNCT_2:def 1
    .= dom r by FUNCT_2:def 1;
  hence thesis by A1,FUNCT_1:2;
end;
