
theorem Th15:
  for n being Ordinal, L being right-distributive non empty
doubleLoopStr, p,q being Series of n,L, a being Element of L holds a * (p + q)
  = a * p + a * q
proof
  let n be Ordinal, L be right-distributive non empty doubleLoopStr, p1,p2
  be Series of n,L, b be Element of L;
  set q1 = b * (p1 + p2), q2 = b * p1 + b * p2;
A1: now
    let x be object;
    assume x in dom q1;
    then reconsider u = x as bag of n;
    q1.u = b * (p1+p2).u by POLYNOM7:def 9
      .= b * (p1.u + p2.u) by POLYNOM1:15
      .= b * p1.u + b * p2.u by VECTSP_1:def 2
      .= (b*p1).u + b * p2.u by POLYNOM7:def 9
      .= (b*p1).u + (b*p2).u by POLYNOM7:def 9
      .= (b*p1 + b*p2).u by POLYNOM1:15;
    hence q1.x = q2.x;
  end;
  dom q1 = Bags n by FUNCT_2:def 1
    .= dom q2 by FUNCT_2:def 1;
  hence thesis by A1,FUNCT_1:2;
end;
