leading correction to a given Yukawa element is suppressed by at least ${\mathcal{O}}(\epsilon ^{2})$ [^8]. With the underlying family symmetry ordering the correction we know that, before symmetry breaking, the operator giving rise to the correction to a given element must transform in the same way under the family symmetry as the leading term. We have just proved that the difference of the Kähler transformations from the identity is at least of $O(\epsilon )$. Furthermore corrections to $Y_{ij}$ after transformations to canonical Kähler with upper triangular matrices come only from $
Y_{kj}$ with $i>k$ and $Y_{kj}<Y_{ij}$. This implies that a new contribution to $Y_{ij}^t$ is subdominant relative to $Y_{ij}$ at least by $O(\epsilon )$ where $\epsilon =<\theta >/M$. As $\theta $ transforms non-trivially under the family symmetry, to maintain the symmetry property of the leading term, this relative correction must be given by a combination of fields which transforms as a singlet, that is at least of the form $\theta \theta ^{\dagger }$ and hence of ${\mathcal{O}}(\epsilon ^{2}).$ This result applies to hierarchical Yukawa structures. For the case that the $(2,3)$ element saturates the bound of Eq. (\[genform\]) it violates the condition of hierarchical Yukawa couplings and our conclusions above 