Proof
Here is the onetoone correspondence. Let Form be the family of 1forms on M (or U) and let Cov be the family of covariant vector fields on M (or U). Define
by
The linearity property of now follows from the distributive laws of arithmetic. We now define the inverse
by
We need to check that this is a covariant vector field; that is, that it transforms in the correct fashion. But, it x and are two charts, then

= 

(if you don't believe this, look at the ambient coordinates)  
= 

(since F is linear) 
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