Let us denote by $i_A$ the canonical injection of A into X$,i_U$ the canonical injection of U into X; let $j_A$ and $j_U$ denote the corresponding injections of Cyl($i_A$) and Cyl($i_U$) into $X\times \mathbf{I}$.

There exists a continuous mapping $r: X\times \mathbf{I}\rightarrow$ Cyl($i_U$) such that $j_U\circ$ $r(x) =x$ for every point $x\in j_A$(Cyl($i_A$)).

Let $U'$ be an open set of $X'$ such that $A\subset X\cap U'\subset U$. By definition of a normal space (TG, IX, p. 41), there exists an open neighbourhood $V'$ of A in $X'$ such that $A\subset V'\subset \overline{V'}\subset U'$ and a continuous function $\varphi ': X'\rightarrow \mathbf{I}$ which is equal to 1 at every point of A and to 0 at every point of $\complement V'$. Put $\varphi =\varphi '|X$ and $V = X\cap V'$. Let us also denote by $\alpha : U\times \mathbf{I}\rightarrow$ Cyl($i_U$) and $\beta : X\rightarrow$ Cyl($i_U$) the canonical mappings (III, p. 238).

Let $r$ be the mapping of $X\times \mathbf{I}$ into Cyl($i_U$) defined by

$\alpha (x,1-(1-t)\varphi (x))$ for $(x, t)\in U\times \mathbf{I}$,

$\beta (x)$ otherwise.

The mapping $r$ is continuous on $U\times \mathbf{I}$. For every point $(x, t)\in U\times \mathbf{I}$ such that $x\notin V$, we have $\varphi (x) = 0$, whence $r(x, t) =\alpha (x,1) =\beta (x)$. It follows that the mappings $r$ and $\beta \circ$ pr$_1$ coincide on $(X-V)\times \mathbf{I}$, which implies that $r$ is continuous on this subspace. Since U and the interior of X-V cover X, the mapping $r$ is continuous.

Let $y$ be a point of Cyl($i_A$) ; put $(x, t) =j_A(y)$. If $x\in$ A, $\varphi (x) = 1$ and $r(x, t) =\alpha (x, t)$, whence $j_U(r(x, t)) = (x, t)$. Otherwise, $t= 1$ and one has $r(x, t) =\alpha (x,1)$ if $x\in U$ and $r(x, t) =\beta (x)$ otherwise ; hence $j_U(r(x, t)) = (x, t)$ in this case. This entails that $j_U\circ r$ maps every point of $j_A$(Cyl($i_A$)) to itself, whence the proposition.

Put $X'= X$ and let $r: X\times \mathbf{I}\rightarrow$ Cyl($i_U$) be the continuous mapping given by Prop. 1. With the notations introduced in the proof of this proposition, there exists a unique continuous mapping F: Cyl($i_U$)$\rightarrow Z$ such that $F(\alpha (x, t)) =\sigma (x, t)$ for $(x, t)\in U\times \mathbf{I}$ and $F(\beta (x)) =f(x)$ for $x\in$ X (III, p. 238, Prop. 4). The mapping $F\circ r: X\times \mathbf{I}\rightarrow$ Z is a homotopy ; its term maps a point $x\in$ X to $F(r(x,1)) = F(\beta (x)) =f(x)$. If $(x, t)\in A\times \mathbf{I}$, $F(r(x, t)) = F(\alpha (x, t)) =\sigma (x, t)$, therefore this homotopy coincides with $\sigma$ on $A\times \mathbf{I}$.

Let F be a continuous mapping of X into Z such that $F|U =f$. Let us choose a homotopy $\sigma : A\times \mathbf{I}\rightarrow Z$ joining $f'$ to $f$. By Cor. 1, there exists a homotopy $\widetilde{\sigma}: X\times \mathbf{I}\rightarrow Z$ with term F which extends $\sigma$. The origin of $\widetilde{\sigma}$ is then a continuous mapping of X into Z which coincides with $f'$ on A.

Since Z is homotopic to a point, the mapping $g$ is homotopic to a constant mapping $f$. The mapping $f$ extends continuously to X ; the assertion therefore follows from Cor. 2.

Let us denote by $i_A$ the canonical injection of A into X, by $\alpha_A: A\times \mathbf{I}\rightarrow$ Cyl($i_A$) and $\beta_A: X\rightarrow$ Cyl($i_A$) the canonical mappings, and also by $j_A:$ Cyl($i_A$)$\rightarrow$ $X\times \mathbf{I}$ the canonical injection. Since A is closed in X, $j_A$ defines a homeomorphism of Cyl($i_A$) onto the closed subspace $(A\times \mathbf{I})\cup (X\times \{1\})$ of $X\times \mathbf{I}$. Let $\widetilde{\sigma}_0$ be the unique mapping of Cyl($i_A$) into V such that $\widetilde{\sigma}_0\circ \alpha_A=\sigma$ and $\widetilde{\sigma}_0\circ \beta_A=f$; it is continuous (III, p. 238, prop. 4).

Since X is paracompact and $\mathbf{I}$ is compact, the space $X\times \mathbf{I}$ is paracompact (TG, I, p. 70, prop. 17), hence normal (TG, IX, p. 49, prop. 4). There therefore exists a continuous mapping $k: X\times \mathbf{I}\rightarrow \mathbf{R}^n$ such that $k\circ j_A=\widetilde{\sigma}_0$ (TG, IX, p. 45, corollary).

The set U of points $x\in$ X such that $k(x, t)\in$ V for all $t\in \mathbf{I}$ is open in X (IV, p. 439, lemma 1). Hence it is a neighbourhood of A. By corollary 1, applied to the mapping $f$ and to the homotopy $k|U\times \mathbf{I}$, there exists a homotopy $\widetilde{\sigma}: X\times \mathbf{I}\rightarrow V$ with initial map $f$ which coincides with $k$, hence with $\sigma$, on $A\times \mathbf{I}$.

The first projection pr$_1: Z\times K\rightarrow Z$ is proper (TG, I, p. 77, cor. 5), hence closed. Consequently, $\complement U =$ pr$_1(\complement W)$ is closed in Z and U is open.

Let Y and Z be topological spaces; let $f: X\times \{1\} \times Y\rightarrow$ $X\times \{1\} \times Z$ be an $X\times \{1\}$-morphism and let $g: U\times \mathbf{I}\times Y\rightarrow U\times \mathbf{I}\times Z$ be a $U\times \mathbf{I}$-morphism which coincides with $f$ on $U\times \{1\} \times Y$.

There then exists an $X\times \mathbf{I}$-morphism $h: X\times \mathbf{I}\times Y\rightarrow X\times \mathbf{I}\times Z$ which coincides with $f$ on $X\times \{1\} \times Y$ and with $g$ on $A\times \mathbf{I}\times Y$.

Moreover, if $f$ and $g$ are homeomorphisms, one can choose a homeomorphism $h$ having the required properties.

Let $U'$ be an open neighbourhood of A in $X'$ such that $U'\cap X\subset U$. Since the space $X'$ is normal, there exists an open neighbourhood $V'$ of A in $X'$ such that $A\subset V'\subset \overline{V'}\subset U'$ (TG, IX, p. 41). Replacing U by $\overline{V'}\cap X$ if necessary, we may thus suppose that U is closed in X. Let us then take up again the notation of Prop. 1 and of its proof; let $r: X\times \mathbf{I}\rightarrow$ Cyl($i_U$) be a continuous mapping such that $j_U\circ r(x) =x$ for every point $x\in j_A$(Cyl($i_A$)).

Put also $f'=$ pr$_3\circ f$ and $g'=$ pr$_3\circ g$. Since $f'$ and $g'$ coincide on $U\times \{1\} \times Y$, there exists a unique mapping

$\varphi :$ Cyl($i_U$)$\times Y\rightarrow$ Cyl($i_U$)$\times Z$

such that $\varphi (\alpha (x, t), y) = (\alpha (x, t), g'(x, t, y))$ for $(x, t, y)\in U\times \mathbf{I}\times Y$ and $\varphi (\beta (x), y) = (\beta (x), f'(x,1, y))$ for $(x, y)\in X\times Y$. Since U is closed in X, the canonical surjection $\pi$ of $(U\times \mathbf{I})\cup X$ onto Cyl($i_U$) is universally strict (III, p. 239, remark 2). Since the mapping $\varphi \circ (\pi \times$ Id$_Y)$ is continuous, the mapping $\varphi$ is continuous. It is therefore a morphism of Cyl($i_U$)-spaces, and even an isomorphism if $f$ and $g$ are homeomorphisms.

Let then $h: X\times \mathbf{I}\times Y\rightarrow X\times \mathbf{I}\times Z$ be the mapping $r^*(\varphi )$ deduced from $\varphi$ by the base change $r$. It is given by $h(x, t, y) =$ $(x, t$, pr$_2(\varphi (r(x, t), y)))$ for $(x, t, y)\in X\times \mathbf{I}\times Y$. It is a morphism of $X\times \mathbf{I}$-spaces, and even an isomorphism if $\varphi$ is one. For $(x, t, y)\in$ $A\times \mathbf{I}\times Y$, one has $r(x, t) = (x, t)$, whence

$h(x, t, y) = (x, t$, pr$_2(\varphi (x, t, y))) = (x, t, g'(x, t, y)) =g(x, t, y)$,

which proves that $h$ coincides with $g$ on $A\times \mathbf{I}\times Y$. Analogously, for $x\in X$ and $y\in Y$, we have $r(x,1) = (x,1)$, hence

$h(x,1, y) = (x,1$, pr$_2(\varphi (x,1, y))) = (x,1, f'(x,1, y)) =f(x,1, y)$

so that $h$ coincides with $f$ on $X\times \{1\} \times Y$. The corollary is thus proved.

Let us first prove two lemmas.

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.

Let $a$ be a point of B; for every point $t$ of $\mathbf{I}$, there exists an open neighbourhood $W_t$ of $t$ in $\mathbf{I}$ and a neighbourhood $V_t$ of $a$ in B such that E is trivializable over $V_t\times W_t$. There then exist an integer $n >0$ and, for every integer $i$ such that 1 $\leqslant i\leqslant n$, a point $t_i$ of $\mathbf{I}$ such that the interval $[\frac{i-1}{n},\frac{i}{n}]$ is contained in $W_{t_i}($III, p. 272, lemma 4). Put $V =\cap_{1\leqslant i\leqslant n}V_{t_i}$. The fibered space E is trivializable over $V\times$ $[\frac{i-1}{n},\frac{i}{n}]$ for every integer $i$ such that $1\leqslant i\leqslant n$. Lemma 3 then follows from Lemma 2 by induction on $n$.

Let us now prove the proposition. By lemma 3, there exists an open covering $(U_j)_{j\in J}$ of B such that, for every $j\in J$, E is trivializable over $U_j\times \mathbf{I}$. Since the space B is paracompact, we may suppose the covering $(U_j)_{j\in J}$ locally finite (TG, IX, p. 49) and choose a covering $(A_j)_{j\in J}$ of B where, for every $j\in J$, the set $A_j$ is closed in B and contained in $U_j$ (TG, IX, p. 49, prop. 4 and p. 48, cor. 1).

For every open subset U of B, let us denote by $\mathscr{F}(U)$ the set of $U\times \mathbf{I}$-isomorphisms of $\overset{-1}{p}(U\times \mathbf{I})$ onto $\overset{-1}{p_{1}}(U)\times \mathbf{I}$ which induce the identical mapping of $\overset{-1}{p}(U\times \{1\})$ onto $\overset{-1}{p_{1}}(U)\times \{1\}$. For every pair $(V,U)$ of open sets of B such that $U\subset V$, let us denote by $r_{UV}:\mathscr{F}(V)\rightarrow \mathscr{F}(U)$ the mapping which, to a $(V\times \mathbf{I})$-isomorphism $g:\overset{-1}{p}(V\times \mathbf{I})\rightarrow \overset{-1}{p_{0}}(V)\times \mathbf{I}$, associates the $(U\times \mathbf{I})$-isomorphism deduced from $g$ by passing to the subspaces. The pair $\mathscr{F}= ((\mathscr{F}(U)),(r_{UV}))$ is a sheaf on B (I, p. 45, example 4). To prove the proposition, it is enough to prove that the sheaf $\mathscr{F}$ is soft (I, p. 64).

Let $j\in J$, let A be a closed subset of $A_j$, let V be an open set in B such that $A\subset V\subset U_j$, and let $g$ be an element of $\mathscr{F}(V)$. There exists an open neighbourhood W of A such that $\overline{W}\subset V$, for a paracompact space is normal (TG, IX, p. 49, prop. 4). We shall prove that there exists an element $g'$ of $\mathscr{F}(U_j)$ such that $g'|W =g|W$. Corollary 2 (I, p. 65) of prop. 6 then implies that the sheaf $\mathscr{F}$ is soft.

Since the $B\times \mathbf{I}$-space E is trivializable over $U_j\times \mathbf{I}$, we may suppose that $\overset{-1}{p}(U_j\times \mathbf{I}) = U_j\times \mathbf{I}\times F$, where F is a topological space. The element $g$ of $\mathscr{F}(V)$ is then a $V\times \mathbf{I}$-isomorphism of $V\times \mathbf{I}\times F$ onto itself which induces the identical mapping of $V\times$ $\{1\}\times F$. Apply cor. 5 of IV, p. 439, to the spaces $X'= X$, $X = U_j$, A = W, U = V, and Y = Z = F, to the mapping $g: V\times \mathbf{I}\times F\rightarrow V\times \mathbf{I}\times F$ and to the identical mapping of $U_j\times \{1\} \times F$. There therefore exists a $U_j\times \mathbf{I}$-isomorphism $g'$ of $U_j\times \mathbf{I}\times F$ onto itself which induces the identical mapping of $U_j\times \{1\} \times F$ and which coincides with $g$ on $\overline{W}\times \mathbf{I}\times F$, and a fortiori on $W\times \mathbf{I}\times F$. Hence the proposition.

Let $\sigma : B'\times \mathbf{I}\rightarrow B$ be a homotopy joining $f_0$ to $f_1$ and let us denote by $E'$ the space over $B'\times \mathbf{I}$ $\sigma^*E$; this is a locally trivial fibre space. Let us denote by $i_0$ and $i_1$ the mappings of $B'$ into $B'\times \mathbf{I}$ given by $x\mapsto (x,0)$ and $x\mapsto (x,1)$. By Cor. 1, the fibre spaces over $B'$ $i^*_0E'$ and $i^*_1E'$ are isomorphic. Since $\sigma \circ i_0=f_0$, the space over $B'$ $i^*_0E'$ is identified with $f_0^*E$; analogously, the space over $B'$ $i^*_1E'$ is identified with $f_1^*E$. Consequently, the fibre spaces over $B'$ $f_0^*E$ and $f_1^*E$ are isomorphic, as was to be proved.

Let $(E', p')$ be the $B'\times \mathbf{I}$-space deduced from $(E, p)$ by base change by the mapping $\sigma : B'\times \mathbf{I}\rightarrow B$. It is a locally trivial fibre space (I, p. 71, Corollary 2), and the mapping $s: B'\times \{0\} \rightarrow E'$ defined by $s((a,0)) = ((a,0),\widetilde{f}(a))$ for $a\in B'$ is a continuous section of $p'$ over $B'\times \{0\}$. By Proposition 2, there exists a continuous section $\widetilde{s}$ of $p'$ extending $s$. The mapping $\widetilde{\sigma}=$ pr$_2\circ \widetilde{s}$ has the required property.

The sheaf $\mathscr{S}$ on B of sections of M is isomorphic to the sheaf on B of isomorphisms of principal fibre spaces from E onto $E'$.

Let us denote by $p,p'$ and $q$ the projections of the B-spaces E, $E'$ and M. For $(x, x')\in E\times_BE'$ and $f\in F$, let $[x, x', f]$ denote the class in M of the element $((x, x'), f)\in (E\times_BE')\times F$. Let $\mathscr{M}$ denote the sheaf on B of isomorphisms of principal fibre spaces from E onto $E'$; its sections over an open set U of B are the isomorphisms of principal fibre spaces from $E_U$ onto $E'_U$.

Let $e$ denote the identity element of G. Let U be an open set of B and let $\varphi : E_U\rightarrow E'_U$ be an isomorphism of principal fibre spaces with group G and base U. The mapping from $E_U$ into $(E\times_BE')\times^{G\times G}F$ defined by $x\mapsto [x, \varphi (x), e]$ is continuous. For $g\in G$ and $x\in E_U$, it sends $x\cdot g$ to $[x\cdot g, \varphi (x\cdot g), e] = [x, \varphi (x), geg^{-1}] = [x, \varphi (x), e]$. There therefore exists a unique continuous mapping $\alpha_U(\varphi ): U\rightarrow M$ such that $\alpha_U(\varphi )(p(x)) = [x, \varphi (x), e]$ for every $x\in E_U$; we have $\alpha_U(\varphi )\in \mathscr{S}(U)$.

It is immediate that the mappings $\alpha_U$ define a morphism of sheaves $\alpha$ from $\mathscr{M}$ into $\mathscr{S}$.

Suppose first that the principal fibre spaces E and $E'$ are both trivializable; choose sections $i: B\rightarrow E$ and $i': B\rightarrow E'$ of them. Then there exists a unique continuous mapping $\theta$ from M into $B\times G$ such that one has

for $b\in B,g\in G,g'\in G$ and $f\in$ F; it is an isomorphism of B-spaces. Let $\varphi$ be an isomorphism of principal fibre spaces from E onto $E'$; there exists a unique continuous mapping $\gamma : B\rightarrow G$ such that $\varphi (i(b)) =i'(b)\cdot \gamma (b)$ for $b\in B$. The image of $\varphi$ under the mapping $\alpha_B$ is the mapping $b\mapsto \theta^{-1}(b, \gamma (b))$ from B into M. This implies that $\alpha_B$ is a bijection.

Consequently, $\alpha_U$ is a bijection for every open set U of B such that the principal fibre spaces $E_U$ and $E'_U$ are trivializable. By corollary 2 of I, p. 55, this implies that $\alpha$ is an isomorphism of sheaves.

Let F be the topological space G endowed with the left operation of the group $G\times G$ given by $(g, g')\cdot f=g'f g^{-1}$. Let $(M, q)$ be the locally trivial fibre space of base $B\times \mathbf{I}$ and fibre type F associated with the principal fibre space $E\times_{B\times\mathbf{I}}(E_1\times \mathbf{I})$ of group $G\times G$. Put $M_1=\overset{-1}{q}(B\times \{1\})$ and $q_1=$ pr$_1\circ p|M_1$; the B-space $(M_1, q_1)$ is identified with the locally trivial fibre space of fibre type F associated with $E_1\times_BE_1$. By lemma 4, where one takes for principal fibre spaces E and $E'$ both equal to $E_1$, the B-space $M_1$ has a section. Since the $B\times \mathbf{I}$-spaces $(M, q)$ and $(M_1\times \mathbf{I}, q_1\times$Id$_{\mathbf{I}})$ are isomorphic (IV, p. 440, prop. 2), the $B\times \mathbf{I}$-space $(M, q)$ has a section, which implies that the principal fibre spaces of group G, E and $E_1\times \mathbf{I}$, are isomorphic.

There exists an open covering $(U_j)_{j\in J}$ of B such that, for every $j\in J$, the fibre space $E_{U_j}$ is trivialisable. Since the space B is paracompact, we may suppose that the covering $(U_j)_{j\in J}$ is locally finite (TG, IX, p. 49) and choose a covering $(A_j)_{j\in J}$ of B where, for every $j\in J$, the set $A_j$ is closed in B and contained in $U_j$ (TG, IX, p. 49, prop. 4 and p. 48, cor. 1).

By I, p. 65, cor. 2 of prop. 6, it is enough, in order to prove the proposition, to establish the following assertion: let U be an open subset of B such that the principal fibre space $(E_U, p_U)$ is trivialisable, let A be a closed subset of B contained in U, let V be an open neighbourhood of A contained in U, and let $f$ be an element of $\mathscr{F}(V)$; then there exists an open neighbourhood W of A in V and an element $f'$ of $\mathscr{F}(U)$ such that $r_{WU}(f') =r_{WV}(f)$. Let us prove this assertion.

Let W be an open subset of B such that $A\subset W\subset \overline{W}\subset V$. Let $s: U\rightarrow E_U$ be a section of $(E_U, p_U)$. Apply Corollary 3 (IV, p. 438) to the space B, the closed subset $\overline{W}$ and the neighbourhood V of $\overline{W}$, and to the mapping $g=f\circ (s|_V)$ of $E_V$ into $E'$. There then exists a continuous mapping $\widetilde{g}: B\rightarrow E'$ which coincides with $g$ on $\overline{W}$. Put $h=\widetilde{g}|_U$. We have $h|_W=g|_W=f\circ (s|_W)$.

The mapping $H: U\times G\rightarrow E'$ defined by $H(x, g) =h(x)\cdot g$ for $(x, g)\in U\times G$ is continuous and compatible with the operations of G in $U\times G$ and $E'$. Put $f'= H\circ s^{-1}$; this is an element of $\mathscr{F}(U)$. For every $x\in W$, the mappings $f$ and $f'$ coincide at the point $s(x)$, hence at every point of $\overset{-1}{p}(x)$, since they are morphisms of principal fibre spaces. The proposition follows.