the (1, 2)-dimensional supermanifold on which our 1$D$ ordinary theory is generalized. The other quantities in the equation (11) are $$\begin{aligned}
&&d\,=\,dt\,\partial_{t} \longrightarrow \tilde d \,=\,dt\,\partial_{t}\,+\,d\theta\,\partial_{\theta}\,+\,
d\bar\theta\,\partial_{\bar\theta},\qquad 
p_{r}(t)\longrightarrow P_{r}(t,\theta,\bar\theta)\\ \nonumber
&&\lambda^{(1)}\,=\,dt\,\lambda(t)\longrightarrow \tilde \lambda^{(1)}\,=\,dt\,\Lambda(t,\theta,\bar\theta)\,+\,
d\theta\,{\bar F}(t,\theta,\bar\theta)\,+\,d{\bar\theta}\,F(t,\theta,\bar\theta),\end{aligned}$$ where the supervariables $\Lambda(t,\theta,\bar\theta),\, F(t,\theta,\bar\theta),\, \bar F(t,\theta,\bar\theta)$ and $P_{r}(t,\theta,\bar\theta)$ have the following expansions along the $(\theta, \bar\theta)$-directions of (1, 2)-dimensional supermanifold \[25\]: $$\begin{aligned}
&&\Lambda(t,\theta,\bar\theta)\,=\,\lambda(t)\,+\,\theta {\bar R(t)}\,+\,{\bar\theta}R(t)\,
+\,i\,\theta\,{\bar\theta}S(t),\\ \nonumber
&&F(t,\theta,\bar\theta)\,=\,C(t)\,+\,i\,\theta\,{\bar B}_{1}(t)\,
+\,i\,\bar\theta B_{1}(t)\,+\,i\,\theta \,\bar\theta \,s(t), \\\nonumber
&&\bar F(t,\theta,\bar\theta)\,=\,\bar C(t)\,+\,i\,\theta\,{\bar B}_{2}(t)\,
+\,i\,\bar\theta B_{2}(t)\,+\,i\,\theta \, \bar\theta \,\bar s(t), \\\nonumber
&&P_{r}(t,\theta,\bar\theta)\,=\,p_{r}(t)\,+\,\theta\,{\bar K}(t),
+\,\bar\theta K(t)\,+\,i\,\theta \,\bar\theta \, L(t).\end{aligned}$$ We note, in the above, that the secondary variables ($R, \bar R, s, \bar s, K, \bar K$) are fermionic and ($S, B_{1}, {\bar B}_{1}, B_{2}, {\bar B}_{2}, L$) are bosonic in nature. It is elementary to verify that, in the limit $\theta\,=\,\bar\theta = 0$, we get back our 1D variables ($\lambda, C, \bar C, p_{r}$) that are present in the Lagrangian (1). The dual horizontality condition (DHC) \[cf. (11)\] leads to the following \[25\] $$\begin{aligned}
{\bar B}_{1}\,=\,B_{2}\,=\,0,\quad s\,=\,\bar s = 0, \quad
B_{1}\,+\,{\bar B}_{2}\,=\,0,\quad {\bar K}\,=\,\dot{\bar R},\quad K = \dot R,\quad L = \dot S.\end{aligned}$$ The above relationships prove that some of the secondary variables are zero and others are interconnected 