Simplified proofs for the Pro-Lie group theorem and the one-parameter subgroup lifting lemma

This note is devoted to the theory of projective limits of finite dimensional Lie groups, as developed in the recent monograph [Hofmann, K. H., and S. A. Morris, "The Lie Theory of Connected Pro-Lie Groups," EMS Publ. House, 2007]. We replace the original, highly non-trivial proof of the One-Parameter Subgroup Lifting Lemma given in the monograph by a shorter and more elementary argument. Furthermore, we shorten (and correct) the proof of the so-called Pro-Lie Group Theorem, which asserts that pro-Lie groups and projective limits of Lie groups coincide.