the action of the subgroup $Aut^{+}(P)$ of all automorphisms of $P$ with $F^{0}_{,0}>0$.

The group $Aut^{+}(P)$ of automorphisms of the bundle contains two subgroups. One is the subgroup $TC$ of the “time change” proper gauge diffeomorphisms $(X^{0}=T;X^{I},I=1,2,3)
\longmapsto (F(T,;X^{J},J=1,2,3),X^{I},I=1,2,3)$ for an arbitrary smooth function $F(X^{I})$ with $F_{,0}>0.$ The other one (denoted $D(B)$) consists of the lifts to the slices $B_{\phi ,t}$ of the manifold $P$ of the orientation preserving diffeomorphisms of the base $B$ (group of such transformations of $B$ is denoted $Diff^{+}(B)$). To lift a diffeomorphism we use (diffeomorphic) projections $\pi _{\phi ,t}=\pi \vert_{B_{\phi ,t}}:B_{\phi ,t}
\longmapsto B$. If $\phi $ is synchronized, lifted diffeomorphisms do not depend on $T$.

Any automorphism of the bundle $\phi \in
Aut^{+}(P)$ generates the time independent diffeomorphism $\phi
_{B}$ of the base $B$, that is element of $Diff^{+}(B).$ Lifting this element to the element of $D(B)$ we represent the group $Aut^{+}(P)$ as the semi-direct product of the normal subgroup $TC(P)$ and the subgroup $D(B)$. Thus, we have proved the first of two following statements

1.  Automorphisms group $Aut^{+}(P)$ is the semidirect product of the subgroup $D(B)\sim
    Diff^{+}(B)$ of orientation preserving diffeomorphisms of the base $B$ and the subgroup $TC$ of proper gauge transformations 