
theorem Th4:
  for I being non degenerated domRing-like associative commutative
Abelian non empty doubleLoopStr for u,v,w being Element of Q.I holds pmult(u,
  pmult(v,w)) = pmult(pmult(u,v),w)
proof
  let I be non degenerated domRing-like associative commutative Abelian non
  empty doubleLoopStr;
  let u,v,w be Element of Q.I;
A1: v`2 <> 0.I by Th2;
  w`2 <> 0.I by Th2;
  then v`2 * w`2 <> 0.I by A1,VECTSP_2:def 1;
  then reconsider t = [v`1 * w`1, v`2 * w`2] as Element of Q.I by Def1;
  u`2 <> 0.I by Th2;
  then u`2 * v`2 <> 0.I by A1,VECTSP_2:def 1;
  then reconsider s = [u`1 * v`1, u`2 * v`2] as Element of Q.I by Def1;
  pmult(u,pmult(v,w)) = [u`1 * (v`1 * w`1), u`2 * t`2]
    .= [u`1 * (v`1 * w`1), u`2 * (v`2 * w`2)]
    .= [(u`1 * v`1) * w`1, u`2 * (v`2 * w`2)] by GROUP_1:def 3
    .= [(u`1 * v`1) * w`1, (u`2 * v`2) * w`2] by GROUP_1:def 3
    .= [s`1 * w`1, (u`2 * v`2) * w`2]
    .= pmult(pmult(u,v),w);
  hence thesis;
end;
