Surfaces are homeomorphic iff are diffeomorphic.

I have read this statement in several places: "Two surfaces are homeomorphic iff are diffeomorphic". I think the nontrivial implication follows in this manner: First, we triangulate the surface and then we smooth the resulting piecewise linear manifold, but I haven't found and well-written proof. Could you possibly give me a reference for the proof? Thanks in advance.

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What about a square and a sphere? Or you assume that both surfaces are smooth? – Ilya Jan 31 '13 at 19:04
Could you give a reference where you've read it? This page has some references for something similar to your question. – anon271828 Jan 31 '13 at 19:11
See this MathOverflow question, which gives a list of references. – David Moews Jan 31 '13 at 19:23
Ilya: Sorry, you're right, I assume smooth surfaces :) – math_failure Jan 31 '13 at 23:49
anon271828, I just read that in lecture notes on the net and google books without proof, because of that I need an strong reference – math_failure Jan 31 '13 at 23:51