I have a question while reading chapter 11 of Bott&Tu, Differential Forms in Algebraic Topology. The book first defines the Euler class of an oriented sphere bundle (a fiber bundle with fiber $S^n$), and then defines the Euler class of an oriented vector bundle as follows:
If $E$ is an oriented vector bundle, the complement $E^0$ of its zero section has the homotopy type of an oriented sphere bundle. The Euler class of $E$ is defined to be that of $E^0$. (p.118)
But how do we know that the Euler class of $E^0$ is well-defined? Is there a unique oriented sphere bundle that has the homotopy type of $E^0$?