proof of modular lawFirst we show C+(B∩A)⊆B∩(C+A):Note that C⊆B,B∩A⊆B, and therefore C+(B∩A)⊆B.Further, C⊆C+A, B∩A⊆C+A, thusC+(B∩A)⊆C+A.Next we show B∩(C+A)⊆C+(B∩A):Let b∈B∩(C+A). Then b=c+a for some c∈C and a∈A.Hence a=b-c, and so a∈B since b∈B and c∈C⊆B.Hence a∈B∩A, so b=c+a∈C+(B∩A).