Suppose $\Gamma\vdash A\vee \Delta$, where as usual $\Gamma$ and $\Delta$ are thought of as sets pf propositions and the turnstyle is for logical consequence, or entailment.

Given the assumption, may one consider the relation between the top line and the bottom line of the sequent $\frac{\Gamma\vdash A, \Delta}{\Gamma\vdash A\vee B, \Delta}$ to be an entailment on a par with - as in, having the same nature as - the relation between the left and the right side of $\Gamma\vdash A\vee \Delta$, or is there something which prohibits such a point of view?