Phasor proof for RC circuits

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I understand that the voltage phasor for a resistor is in phase with the current phasor in a simple ac circuit consisting of only an AC voltage source and the resistor.

I understand that the voltage phasor for a capacitor is pi/2 radians behind the current phasor for an AC circuit consisting of only an AC voltage source and a capacitor.

When an RC circuit consisting of a resistor and capacitor in series is analyzed, I don't understand how the above still holds. Both proofs were based on the fact that the voltage across each circuit element varies by E = E0*cos(wt), but now the sum of the voltages across each element vary by this NOT each individual voltage.

I feel like this is a pure assumption. Here are the links to the book proof. I have a problem with the second picture of the phasor diagrams.
http://imgur.com/wmjzJ6C
http://imgur.com/aeh1ESb
 

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