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186 P. Carrillo Rousein the following way. For x 2 V we have .x/ 2 R p f0g. If we write .x/ D. 1 .x/; 0/, then 1 W V ! V R pis a diffeomorphism, where V D U \.R p f0g/. Set Q.v;;0/ D . 1 .v/; d N v ./; 0/and Q.u; t/ D ..u/; t/ for t ¤ 0. Here d N v W N v ! R q is the normal componentof the derivative d v for v 2 V. It is clear that Q is also a bijection (in particular itinduces a C 1 structure with border over D U V ).Let us define, with the same notations as above, the following set: U V Df.x;;t/2 Rp R q Œ0; 1 W .x; t / 2 U g:This is an open subset of R p R q Œ0; 1 and thus a C 1 manifold (with border). It isimmediate that DV U is diffeomorphic to U Vthrough the restriction of ‰ given in (3).Now we consider an atlas f.U˛;˛/g˛2 of M consisting of X-slices. It is clear thatD M X D [˛2 D U˛V˛(4)and if we take D U˛V˛'˛! U˛ defined as the compositionV˛D U˛V˛then we obtain the following result.˛! D U˛V˛‰ 1˛! U˛V˛Proposition 3.1. f.D U˛ ;'˛/g˛2 is a C 1 atlas with border over D M V˛ X .In fact the proposition can be proved directly from the following elementary lemma.Lemma 3.2. Let F W U ! U 0 be a C 1 diffeomorphism where U R p R q andU 0 R p R q are open subsets. We write F D .F 1 ;F 2 / and we suppose thatF 2 .x; 0/ D 0. Then the function FQW U V ! U 0Q F.x;;t/DVdefined by0´.F1 .x; 0/; @F 2.x; 0/ ;0/@if t D 0;.F 1 .x; t/; 1 t F 2.x;t/;t/ if t>0:is a C 1 map.Proof. Since the result will hold if and only if it is true in each coordinate, it is enoughto prove that if we have a C 1 map F W U ! R with F.x;0/ D 0, then the mapQF W U V ! R given by Q F.x;;t/Dis a C 1 map. For that we write´@F@1t.x; 0/ if t D 0;F.x;t/ if t>0F.x;/ D @F .x; 0/ C h.x; / @