
theorem Th3:
  for I being non degenerated domRing-like associative commutative
Abelian add-associative distributive non empty doubleLoopStr for u,v,w being
Element of Q.I holds padd(u,padd(v,w)) = padd(padd(u,v),w)
proof
  let I be non degenerated domRing-like associative add-associative Abelian
  distributive commutative non empty doubleLoopStr;
  let u,v,w be Element of Q.I;
A1: u`1 * (v`2 * w`2) + (v`1 * w`2 + w`1 * v`2) * u`2 = u`1 * (v`2 * w`2) +
  ((v`1 * w`2) * u`2 + (w`1 * v`2) * u`2) by VECTSP_1:def 3
    .= (u`1 * (v`2 * w`2) + (v`1 * w`2) * u`2) + (w`1 * v`2) * u`2 by
RLVECT_1:def 3
    .= (u`1 * (v`2 * w`2) + (v`1 * w`2) * u`2) + w`1 * (v`2 * u`2) by
GROUP_1:def 3
    .= ((u`1 * v`2) * w`2 + (v`1 * w`2) * u`2) + w`1 * (v`2 * u`2) by
GROUP_1:def 3
    .= ((u`1 * v`2) * w`2 + (v`1 * u`2) * w`2) + w`1 * (v`2 * u`2) by
GROUP_1:def 3
    .= (((u`1 * v`2) + (v`1 * u`2)) * w`2) + w`1 * (v`2 * u`2) by
VECTSP_1:def 3;
A2: v`2 <> 0.I by Th2;
  u`2 <> 0.I by Th2;
  then u`2 * v`2 <> 0.I by A2,VECTSP_2:def 1;
  then reconsider s = [u`1 * v`2 + v`1 * u`2, u`2 * v`2] as Element of Q.I by
Def1;
  w`2 <> 0.I by Th2;
  then v`2 * w`2 <> 0.I by A2,VECTSP_2:def 1;
  then reconsider t = [v`1 * w`2 + w`1 * v`2, v`2 * w`2] as Element of Q.I by
Def1;
  padd(u,padd(v,w)) = [u`1 * t`2 + (v`1 * w`2 + w`1 * v`2) * u`2, u`2 * t`2]
    .= [u`1 * (v`2 * w`2) + (v`1 * w`2 + w`1 * v`2) * u`2, u`2 * t`2]
    .= [u`1 * (v`2 * w`2) + (v`1 * w`2 + w`1 * v`2) * u`2, u`2 * (v`2 * w`2)
  ]
    .= [u`1 * (v`2 * w`2) + (v`1 * w`2 + w`1 * v`2) * u`2, (u`2 * v`2) * w`2
  ] by GROUP_1:def 3;
  then padd(u,padd(v,w)) = [s`1 * w`2 + w`1 * (v`2 * u`2), (u`2 * v`2) * w`2]
  by A1
    .= [s`1 * w`2 + w`1 * s`2, (u`2 * v`2) * w`2]
    .= padd(padd(u,v),w);
  hence thesis;
end;
