hopf algebras - Action of Co-quasi-triangular Universal r-form on $a otimes 1$

Like David said, the proof is almost identical to the earlier one for $R$-matrices:

$r(xotimes 1) = rcirc (idotimesmu)(xotimes 1otimes 1) = (r_{13}ast r_{12})(xotimes 1otimes1)= sum r(x'otimes 1)r(x''otimes 1) = (r ast r)(xotimes 1).$

Since $r$ is invertible, $r(xotimes 1)=epsilon(x)$.