Let's play algebra.

{(m/(n+m))*(V+mK) - mK} =? {(m/(n+m))*(V-nK)}

Notice that [m / (m+n)] is common to both expressions. So we eliminate that to simplify the two expressions. Divide both sides by [m / (m+n)],

(V +mK) -(mK / [m /(m+n)]) =? V -nK

Then simplify. Expand the Lefthand Side,

(V +mK) -(mK * [(m+n) / m]) =? V -nK

(V +mK) -[(mK)(m+n) / m] =? V -nK

Clear the fraction, multiply both sides by m,

m(V +mK) -mK(m+n) =? mV -mnK

mV +(m^2)K -(m^2)K -mnK =? mV -mnK

mV -mnK =? mV -mnK

Yes.

So, it is shown.