finiteness of $C_{\cup }$ and $C_{\cap }$

Theorem 1

For any set $L$ and any subsets $X$ and $Y$ of $L$, the consequenceoperator $C_{\mathrm{\cap }}\mathit{}\mathrm{(}X\mathrm{,}Y\mathrm{)}$ is finitary.

Theorem 2

For any set $L$, any subsets $X$ of $L$, and any finite subset $Y$of $L$, the consequence operator $C_{\mathrm{\cap }}\mathit{}\mathrm{(}X\mathrm{,}Y\mathrm{)}$ is finitary.