theorem
  for K being associative left_unital Abelian
     add-associative right_zeroed right_complementable non empty doubleLoopStr
  for a1,a2,b1,b2 being Element of K holds
    <*a1,a2*> "*" <*b1,b2*> = a1*b1 + a2*b2
proof
  let K be associative left_unital Abelian add-associative
  right_zeroed right_complementable non empty doubleLoopStr;
  let a1,a2,b1,b2 be Element of K;
  set p=<*a1,a2*>;
  set q=<*b1,b2*>;
  (the addF of K)$$(mlt(p,q))=(the addF of K)$$<*a1*b1,a2*b2*> by Lm5
    .=a1*b1 + a2*b2 by FINSOP_1:12;
  hence thesis;
end;
