reserve a,b,c,d,e,f for Real,
        k,m for Nat,
        D for non empty set,
        V for non trivial RealLinearSpace,
        u,v,w for Element of V,
        p,q,r for Element of ProjectiveSpace(V);

theorem Th6:
  for K being commutative associative left_unital Abelian
  add-associative right_zeroed right_complementable non empty doubleLoopStr for
  a1,a2,a3,b1,b2,b3 being Element of K holds
  <*a1,a2,a3*> "*" <*b1,b2,b3*> = a1*b1 + a2*b2 + a3*b3
  proof
    let K be commutative associative left_unital Abelian add-associative
    right_zeroed right_complementable non empty doubleLoopStr;
    let a1,a2,a3,b1,b2,b3 be Element of K;
    set p = <*a1,a2,a3*>, q = <*b1,b2,b3*>;
    Sum(mlt(p,q)) = (the addF of K) $$ mlt(p,q) by FVSUM_1:def 8
                 .= (the addF of K) $$ <*a1*b1,a2*b2,a3*b3*> by Th5
                 .= a1*b1 + a2*b2 + a3*b3 by FINSOP_1:14;
    hence thesis by FVSUM_1:def 9;
  end;
