l. h. s. of relation (21) has been already computed clearly by taking the help of the Hodge duality operation $\star$ defined on the (2, 2)-dimensional supermanifold \[22\].

At this stage, we would like to clarify some of the [*new*]{} symbols used in equation (21). We have the generalization of the ordinary exterior derivative $d = dx^{\mu}\,\partial_{\mu}$ and Abelian 1-form $A^{(1)} = dx^{\mu}\,A_{\mu}$ onto the (2, 2)-dimensional supermanifold as $$\begin{aligned}
&&d\,=\,dx^{\mu}\,\partial_{\mu}\rightarrow\,\tilde{d}\,=\,dx^{\mu}\,\partial_{\mu}\,+
\,d\theta\,\partial_{\theta}\,+\,d\bar{\theta}\,\partial_{\bar{\theta}},\nonumber\\
&&A^{(1)}\,=\,dx^{\mu}\,A_{\mu}\,\rightarrow\,\tilde{A}^{(1)}\,
=\,dx^{\mu}\,{\cal B}_{\mu}(x,\, \theta, \bar{\theta})\,+\,d\theta\,\bar{F}(x, \theta, \bar{\theta})\,
+\,d\bar{\theta}\,F(x, \theta, \bar{\theta}),\end{aligned}$$ where the superfields ${\cal B}_{\mu}(x, \theta, \bar\theta), F(x, \theta, \bar\theta)$ and $\bar F (x, \theta, \bar\theta)$ have the following expansions along ($\theta, \bar\theta$)-directions of the (2, 2)-dimensional supermanifold $$\begin{aligned}
&&{\cal B}_{\mu}(x, \theta , \bar{\theta})\, = \,A_{\mu}(x)\,+
\,\theta\,R_{\mu}^{(1)}(x)\,+\,\bar{\theta}\,R_{\mu}^{(2)}(x)\,
+\,i\,\theta\,\bar{\theta}\,S_{\mu}(x),\nonumber\\
&&F(x, \theta , \bar{\theta})\, = \,C(x)\,+\,i\,\theta\, B_{1}(x)\,+\,i\,\bar{\theta}\,B_{2}(x) \, 
+\,i\,\theta\,\bar{\theta}\,s(x),\nonumber\\
&&\bar{F}(x, \theta , \bar{\theta})\,= \,\bar{C}(x)\,+\,i\,\theta\, B_{3}(x)\,+
\,i\,\bar{\theta}\,B_{4}(x)+\,i\,\theta\,\bar{\theta}\,\bar s(x),\end{aligned}$$ where ($A_{\mu}(x), C(x), \bar C(x)$) are the basic fields of the modified versions of the 2$D$ [*Proca*]{} and [*anomalous gauge theories*]{}. The set of secondary fields ($R_{\mu}^{(1)}, R_{\mu}^{(2)}, s, \bar s$) are fermionic and ($S_{\mu}, B_{1}, B_{2}, B_{3}, B_{4}$) are bosonic in nature (because of the fermionic nature of the Grassmannian variable ($\theta, \bar\theta$)). The dual horizontality condition (21) leads to the following 