Let $(P,\pi,M,G)$ be a principal fibration, $A$ a principal connection on $P$ (i.e. $\forall p \in P, T_pP = A_p \oplus V_p$), $\omega$ the connection 1-form of $A$, $f$ a gauge transformation of $P$, $A \cdot f = f^{-1}(A)$ the right action of $Gauge(P)$ the gauge group on the affine space of principal connections on $P$, $\omega^f$ the connection 1-form of $f^{-1}(A)$.
In Andrei Teleman's book "introduction to Gauge Theory" (in french), there is the following formula (without any proof) for the effect of a gauge transformation of the connection 1-form of a principal connection :
$\omega^f = f^* \omega = \omega + \theta^{-1}(d^A \theta)$, where :
$\theta : P \to G$ corresponds to $f$ via the isomorphism $Gauge(P) \simeq \Omega^{0}_{Int}(P,G)$ (this last space being the $Int$-equivariant applications $P \to G$) and $\theta$ being defined by $\forall p \in P, f(p) = p \cdot \theta(p)$,
$d^A \theta$ is the exterior covariant differentiation associated to $A$ (i.e. $d^A \theta(X) = T_p \theta(pr^A X)$ for $x \in T_pP$,
and the 1-form $\theta^{-1}(d^A \theta) \in \Omega^{0}_{Ad}(P,\mathfrak{g})$ is defined for $p \in P$ and $X \in T_pP$ by $(\theta^{-1}(d^A \theta)_p (X) = T_{\theta(p)} L_{\theta(p)^{-1}} \circ T_p \theta (pr^A X) = \omega_{mc}(T_p \theta (pr^A X))$, where $\omega_{mc}$ is the Maurer-Cartan 1-form on $G$.
$\omega^f = f^* \omega$ is OK, and I can demonstrate the whole formula for vertical tangent vector $X \in V_pP$. But I can't for $X$ horizontal.
I think that this formula could even be proved by supposing $P$ trivial, i.e. $P = U \times G$, with $U$ open set of $M$, and the connection $A$ trivial. But even in this case I can't prove the whole formula.
And I can't find either books or pdf with a clean proof.
Any help ?