
theorem Th8:
  for L be associative non empty doubleLoopStr
  for a,b being Element of L, p being sequence of L holds (a*b)*p = a*(b*p)
proof
  let L be associative non empty doubleLoopStr;
  let a,b be Element of L, p be sequence of L;
  for i be Element of NAT holds ((a*b)*p).i = (a*(b*p)).i
  proof
    let i be Element of NAT;
    thus ((a*b)*p).i = (a*b)*p.i by POLYNOM5:def 4
      .= a*(b*(p.i)) by GROUP_1:def 3
      .= a*(b*p).i by POLYNOM5:def 4
      .= (a*(b*p)).i by POLYNOM5:def 4;
  end;
  hence thesis by FUNCT_2:63;
end;
