Corollary 1

Let X and Y be topological spaces and let $f: X\rightarrow Y$ be a continuous relatively connected mapping. Let T and $T'$ be coverings of Y. If the coverings $X\times_YT$ and $X\times_YT'$ of X are isomorphic, the coverings T and $T'$ are isomorphic.

Let in fact $h: X\times_YT\rightarrow X\times_YT'$ and $h': X\times_YT'\rightarrow X\times_YT$ be X-isomorphisms inverse to one another. By proposition 5, there exist Y-morphisms $g: T\rightarrow T'$ and $g': T'\rightarrow T$ such that $f^*(g) =h$ and $f^*(g') =h'$. We then have $f^*(g'\circ g) =f^*(g')\circ f^*(g) =$ Id$_{X\times_YT}$, therefore $g'\circ g=$ Id$_T$, because the mapping $f^*$ is injective. Analogously, $g\circ g'=$ Id$_{T'}$. The coverings T and $T'$ are therefore isomorphic.