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Oct 22 at 16:32 comment added Thorgott Apologies for the late reply and not having mentioned this in the post, but I am familiar with Whitehead's excellent book. The proof there, however, uses simplicial approximation and homological methods, so I don't really think it is in the homotopical spirit as the proof indicated by May or Strom that I'm looking for.
Oct 7 at 15:31 history answered John Rognes CC BY-SA 4.0