
theorem Th10:
  for X being set, L being left-distributive non empty
  doubleLoopStr, p being Series of X,L, a,a9 being Element of L holds
  a * p + a9 * p = (a + a9) * p
proof
  let n be set, L be left-distributive non empty doubleLoopStr, p be Series
  of n,L, a,a9 be Element of L;
  set p1 = a * p + a9 * p, p2 = (a + a9) * p;
A1: now
    let u be object;
    assume u in dom p1;
    then reconsider u9 = u as bag of n;
    p1.u9 = (a*p).u9 + (a9*p).u9 by POLYNOM1:15
      .= a * p.u9 + (a9*p).u9 by POLYNOM7:def 9
      .= a * p.u9 + a9 * p.u9 by POLYNOM7:def 9
      .= (a + a9) * p.u9 by VECTSP_1:def 3
      .= p2.u9 by POLYNOM7:def 9;
    hence p1.u = p2.u;
  end;
  dom p1 = Bags n by FUNCT_2:def 1
    .= dom p2 by FUNCT_2:def 1;
  hence thesis by A1,FUNCT_1:2;
end;
