Is my proof of the following statement correct? Correctness and completeness have been proven for L and can be used in the proof.
If $\Sigma \vdash_L H_1 \lor H_2 $ then $\Sigma \vdash_L H_1 $ or $\Sigma \vdash_L H_2 $.
Proof: Let v be a model of $\Sigma$. Now assume that $\Sigma \vdash_L H_1 \lor H_2 $. By correctness of L it follows that $\Sigma \vDash H_1 \lor H_2 $, so v is also a model of $H_1 \lor H_2 $. However, the truth function of $\lor$ states that $f(H_1 \lor H_2, v)=T$ iff $f(H_1)=T$ or $f(H_2)=T$. So it follows that $\Sigma \vDash H_1$ or $\Sigma \vDash H_2 $. By completeness of L it follows that $\Sigma \vdash_L H_1 $ or $\Sigma \vdash_L H_2 $. ∎