Lemma 2

Let $\alpha ,\beta ,\gamma$ be real numbers such that $\alpha \leqslant \beta \leqslant \gamma$, let B be a topological space and let $p: E\rightarrow B\times [\alpha , \gamma ]$ be a continuous mapping. Put $B_0= B\times [\alpha , \beta ]$, $B_1= B\times [\beta , \gamma ]$, $E_0=\overset{-1}{p}(B_0)$, $E_1=$ $\overset{-1}{p}(B_1)$ and denote by $p_0: E_0\rightarrow B_0,p_1: E_1\rightarrow B_1$ the mappings deduced from $p$. If $(E_0, p_0)$ and $(E_1, p_1)$ are trivializable fibre spaces, the same is true of $(E, p)$.

Let $g_0: E_0\rightarrow B_0\times F_0$ and $g_1: E_1\rightarrow B_1\times F_1$ be trivializations of $E_0$ and $E_1$ respectively. Let $g'_0$ and $g'_1$ denote the trivializations of the $B\times \{\beta \}$-fibered space $\overset{-1}{p}(B\times \{\beta \})$ deduced from $g_0$ and $g_1$ by restriction. The mapping $h=g'_0\circ (g'_1)^{-1}$ is a $B\times \{\beta \}$-isomorphism of $B\times \{\beta \} \times F_1$ onto $B\times \{\beta \} \times F_0$. Define a continuous mapping $h'$ of $B\times F_1$ into $F_0$ by putting $h'(a, y) =$ pr$_3\circ h(a, \beta , y)$ for $(a, y)\in B\times F_1$. For $(a, t, y)\in B\times [\beta , \gamma ]\times F_1$, put $H(a, t, y) = (a, t, h'(a, y))$. The mapping H resulting is a $(B\times [\beta , \gamma ])$-isomorphism of $B\times [\beta , \gamma ]\times F_1$ onto $B\times [\beta , \gamma ]\times F_0$, and one has $g_0|\overset{-1}{p}(B\times \{\beta \}) = H\circ g_1|\overset{-1}{p}(B\times \{\beta \})$. There therefore exists a continuous mapping $g: E\rightarrow B\times [\alpha , \gamma ]\times F_0$ such that $g|E_0=g_0$ and $g|E_1= H\circ g_1$. The mapping $g$ is an isomorphism of $B\times [\alpha , \gamma ]$-spaces, hence E is trivializable.