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

Given the assumption, may one consider the relation between what is above the line and what is below the 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 the turnstyle in $\Gamma\vdash A\vee \Delta$, or is there something which prohibits such a point of view?