Theta discretization PDE

Question

I am trying to understand the validity of why we can theta discretize the solution to a PDE. For a PDE following: $$0 = \partial_tf + A f$$ I understand that for one discrete time step the solution to above would be $$f(t_h) = e^{\Delta t A} f(t_{h+1}) = \sum_{k=0}^\infty \frac{(\Delta t A)^2}{k!} f(t_{h+1}),$$ using a Taylor expansion and $\Delta t =t_{h+1} - t_h$. Then my question is, how can we see that the theta scheme solves above: \begin{align} f(t_{h+1/2}) &=[I+(1-\theta)\Delta t A] f(t_{h+1})\\ [I-\theta\Delta t A]f(t_h) &= f(t_{h+1/2}). \end{align}