= Transition function of a principal bundle
{title2=$\Phi_j\Phi_i^{-1}(x,g)=(x,\psi_{ji}(x)g)$}
= Transition functions of a principal bundle
{synonym}
For equivariant <local trivializations> of a right <principal bundle>, the change of coordinates is left multiplication in the group coordinate. Indeed an equivariant change $H$ satisfies $H(x,gh)=H(x,g)h$, so evaluating at the identity gives $H(x,g)=(x,\psi(x)g)$. Its inverse shows that $\psi$ takes values in the <Lie group>, and restriction to $(x,e)$ makes it smooth. Composition on triple overlaps gives $\psi_{ki}=\psi_{kj}\psi_{ji}$.
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