As a property of equations, you can add or subtract the same quantity to/from both sides, and the truth value of the equality will remain the same: if you start with something true, e.g. 7=7 and add the same quantity to both sides, e.g. 7+4=7+4, you will still have something true, i.e. 11=11.
On the other hand, if you start with something false, e.g. 3=7 and add the same quantity to both sides, e.g. 3+6=7+6, you will still have something false, i.e. 9=13.
So, you can subtract [tex] m_B [/tex] from both sides to transform the equality into
[tex] m_A+ m_B=m_B + m_C \to m_A+ m_B-m_B=m_B + m_C-m_B \to m_A=m_C [/tex]
as required.
