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@Troy_S is correct on his comments I will elaborate a bit.
If you are adding an element with alpha channel over the hole of the same element on the background, created by another alpha channel, you will get those edges. The images are being added but the alpha channels are not combined.
Only the elements being overlaid should have alpha not the background.
The idea is that you are adding an element over another using the alpha channel of the element on top, not one on top of a hole in the underlaying image. The proper operation is Alpha Over.
If you must "add" images on top of the alpha channel of another image (i don't know why you would), then all alphas should be combined.
Add the RGB image and the ALPHA channels separately and associate the alpha to the RGB information with an add alpha node.
Note however that adding is not the right concept to use when layering images. Alpha Over uses the alpha channel to control the mix of the top element. Add will not use alpha, it just adds the RGB values of one image on top of the values of another, and you might get into situations where many layers added together might look semi-transparent and bright instead of on top of each other.
To deal with more than two layers just use more alpha over nodes and add all of it on top of the background, like in this example:
For more info:
Cycles alpha mask hole