$g(x)=\frac{L}{2}\|x\|_2^2-f(x)$, is a convex function. Therefore by first-order condition, there is
\begin{align*} g(y) \geq g(x) + (Lx-\nabla f(x))^T(y-x) \,\,\,\,(1)\\ g(x) \geq g(y) + (Ly-\nabla f(y))^T(x-y) \,\,\,\,(2) \end{align*}
By adding these two inequalities, I got,
$(Lx-\nabla f(x))^T(y-x)+ (Ly-\nabla f(y))^T(x-y) \leq 0 \, (3)$
My question is, it is always easy to construct two inequalities in the same form and by adding them together, we can get interesting structures. However, is there anyway that I can prove back? That is, given (3), can I derive (1) or (2)?