proof of modular law

First we show $C+(B\cap A)\subseteq B\cap (C+A)$:
Note that $C\subseteq B,B\cap A\subseteq B$, and therefore $C+(B\cap A)\subseteq B$.
Further, $C\subseteq C+A$, $B\cap A\subseteq C+A$, thus $C+(B\cap A)\subseteq C+A$.

Next we show $B\cap (C+A)\subseteq C+(B\cap A)$:
Let $b\in B\cap (C+A)$. Then $b=c+a$ for some $c\in C$ and $a\in A$. Hence $a=b-c$, and so $a\in B$ since $b\in B$ and $c\in C\subseteq B$.
Hence $a\in B\cap A$, so $b=c+a\in C+(B\cap A)$.

Title	proof of modular law
Canonical name	ProofOfModularLaw
Date of creation	2013-03-22 12:50:45
Last modified on	2013-03-22 12:50:45
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	8
Author	yark (2760)
Entry type	Proof
Classification	msc 16D10
Related topic	FirstIsomorphismTheorem