
theorem Th14:
  for L being associative non empty doubleLoopStr for p being
Polynomial of L for x1, x2 being Element of L holds x1 * (x2 * p) = (x1 * x2) *
  p
proof
  let L be associative non empty doubleLoopStr, p being Polynomial of L;
  let x1, x2 be Element of L;
  set f = x1 * (x2 * p), g = (x1 * 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*(x2*p.i) by POLYNOM5:def 4
      .= (x1*x2)*p.i by GROUP_1:def 3
      .= g.i by POLYNOM5:def 4;
    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;
