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The purpose of this paper is to prove equivariant versions of some basic theorems in differential topology for proper Lie group actions. In particular, we study how to extend equivariant isotopies and then apply these results to obtain equivariant smoothing and gluing theorems. We also study equivariant collars and tubular neighbourhoods. When possible, we follow the ideas in the well-known book of M W Hirsch. When necessary, we use results from the differential topology of Hilbert spaces. 57S20doi:10.2140/agt.2007.7.1 fatcat:4yg4rcb76ze7jmhuvgmrp4ca3y