I am self-studying https://sayanmuk.github.io/StochasticAnalysisManifolds.pdf and I am struggling with the proof of the following proposition:
Proposition 2.4.2: Let $\theta$ be a 1-form on $M$ and $X$ the solution of the equation $dX_t = V_\alpha(X_t) \circ dZ_\alpha$. Then, $$ \int_{X[0,t]} \theta = \int_{0}^{t} \theta(V_\alpha)(X_s) \circ dZ_\alpha $$
Proof: From Lemma 2.3.8, we have $dW_t = U_t^{-1} V_{\alpha}(X_t) \circ dZ_{\alpha}$. Hence, the differential of the line integral is: $$\widetilde{\theta}(U_t) \circ dW_t = \left\langle \widetilde{\theta}(U_t), U_t^{-1} V_{\alpha}(X_t) \right\rangle \circ dZ_{\alpha} = \theta(V_{\alpha})(X_t) \circ dZ_{\alpha} $$
My problem is understanding how the Stratonovich integral was applied, or more specifically the operator $\circ$.
By definition, the Stratonovich integral relates with the Ito integral in the following way: $X_t = X_0 + \int_{0}^{t} V_\alpha(X_s) \, dZ_\alpha(s) + \frac{1}{2} \int_{0}^{t} \nabla_{V_\beta} V_\alpha(X_s) \, d\langle Z_\alpha \, dZ_\beta\rangle_s$.
I could see the second term with the covariation is very similar apart from the differential to $\langle\tilde{\theta} (U_t), U^{-1}_t V_\alpha(X_t) \rangle$, but even so it needs to be differentiated. The first component I do not know how it would disappear. This is just a summary of my handwritten attempts.
Question: Can someone explain me this first step in the proof?