How to derive ${ A vdash C }$ from ${A lor B vdash C}$ in the sequent calculus LK?

From axiom and $lor$-r: $cfrac { A vdash A}{ A vdash A lor B }$ Assumption: $A lor B vdash C$. Then apply Cut to get $Avdash C$.

Without $text {Cut}$ [and without proper tree formatting...]:

1. right branch:

$cfrac { C vdash C }{ A, C vdash C }$, by Ax and Weak, then:

$C vdash A supset C$, by $supset text {-r}$.

2. left branch:

$cfrac { A vdash A }{ A vdash A, C }$, by Ax and Weak, then:

$A vdash A lor B, C$, by $lor text {-r}$, then:

$vdash A lor B, A supset C$, by $supset text {-r}$.

Finally:

$cfrac { vdash A lor B, A supset C qquad C vdash A supset C }{ A lor B supset C vdash A supset C }$, by $supset text {-l}$.

Finally:

$vdash (A lor B supset C) supset (A supset C)$, by $supset text {-r}$.