Why do voltage gradient and electric field direction correlate?

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Why do the greatest voltage and the direction of electric field lines correlate with one another? Is there a mathematical proof for this?
 

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To keep things simple, suppose the electric field is uniform, otherwise we have to use calculus. The potential difference between two points is defined as the negative of the work done by the electric force on a charge that moves between those points:

[tex]\Delta V = -W_e = -\vec F_e \cdot \vec{\Delta x} = -q \vec E \cdot \vec{\Delta x}= - qE \Delta x \cos \theta[/tex]

where theta is the angle between the electric field and the displacement of the charge. Which angle makes the potential difference a maximum?