On page 67 here -> http://www1.spms.ntu.edu.sg/~frederique/AA10.pdf

I understand why InJ is a subset of IJ right up to ax + ay is an element of IJ.

I figured that ax + ay = yx + xy (since a is in I and a is in J) so we have xy and yx are in IJ... but how do we know that when you add two elements of IJ, you still get an element of IJ... like, how do we know xy + xy is an element of IJ?