electric field between plates of capacotior

There is a no set detailed proof of the fact that electric fields between plates of a parallel plate capacitor is non-zero, while outside it is zero. And this can be shown by simply using the very basics of electric field like shown below. 

 

The Electric Field due to a sheet of charged infinite sheet is very simple in that the E-field is constant at any distance from the sheet.

When two plate of different charge are placed near each other, the two E-fields between the plates add while the E-field outside the plate cancel.