There are different ways to derive the summation theorems. One is analytical and rigorous using a combination of linear algebra and calculus. The other is less rigorous, but more operational and intuitive. The latter derivation is shown here.
Consider the two-step pathway:
where  and  are fixed species so that the system can achieve a steady-state.
Let the pathway be at steady-state and imagine increasing the concentration of enzyme, , catalyzing the first step, , by an amount, . The effect of this is to increase the steady-state levels of S and flux, J. Let us now increase the level of  by  such that the change in S is restored to the original value it had at steady-state.
The net effect of these two changes is by definition, .
There are two ways to look at this thought experiment, from the perspective of the system and from the perspective of local changes. For the system we can compute the overall change in flux or species concentration by adding the two control coefficient terms, thus:
We can also look at what is happening locally at every reaction step for which there will be two: one for , and another for . Since the thought experiment guarantees that , the local equations are quite simple:
where the  terms are the elasticities. However, because the enzyme elasticity is equal to one, these reduce to:
Because the pathway is linear, at steady-state, . We can substitute these expressions into the system equations to give:
Note that at steady state the change in  and  must be the same, therefore .
Setting , we can rewrite the above equations as:
We then conclude through cancelation of  since , that: