reserve i,j,k for Nat;

theorem Th10:
  for K being distributive non empty doubleLoopStr holds
  the multF of K is_distributive_wrt the addF of K
proof
  let K be distributive non empty doubleLoopStr;
  now
    let a1,a2,a3 be Element of K;
    thus (the multF of K).(a1,(the addF of K).(a2,a3)) = a1*(a2+a3)
      .= a1*a2+a1*a3 by VECTSP_1:def 7
      .= (the addF of K).((the multF of K).(a1,a2),(the multF of K).(a1,a3));
    thus (the multF of K).((the addF of K).(a1,a2),a3) = (a1+a2)*a3
      .= a1*a3+a2*a3 by VECTSP_1:def 7
      .= (the addF of K).((the multF of K).(a1,a3),(the multF of K).(a2,a3));
  end;
  hence thesis by BINOP_1:11;
end;
