
theorem Th17:
  for L being left-distributive non empty doubleLoopStr for p
being Polynomial of L for x1, x2 being Element of L holds (x1 + x2) * p = x1 *
  p + x2 * p
proof
  let L be left-distributive non empty doubleLoopStr, p be Polynomial of L;
  let x1,x2 be Element of L;
  set f = (x1 + x2) * p, g = x1 * p + x2 * p;
A1: now
    let i9 be object;
    assume i9 in dom f;
    then reconsider i = i9 as Element of NAT;
    f.i = (x1+x2)*p.i by POLYNOM5:def 4
      .= x1*p.i+x2*p.i by VECTSP_1:def 3
      .= (x1*p).i+x2*p.i by POLYNOM5:def 4
      .= (x1*p).i+(x2*p).i by POLYNOM5:def 4
      .= g.i by NORMSP_1:def 2;
    hence f.i9 = g.i9;
  end;
  dom f = NAT by FUNCT_2:def 1
    .= dom g by FUNCT_2:def 1;
  hence thesis by A1,FUNCT_1:2;
end;
