$\left.\begin{array}{rr} \mathcal{B}_1, \,\ \tau_1 \text{ için baz}\Rightarrow \mathcal{B}_1\subseteq \tau_1\\ \\ \mathcal{B}_2, \,\ \tau_2 \text{ için baz}\Rightarrow \mathcal{B}_2\subseteq \tau_2 \end{array}\right\}\Rightarrow $
$\Rightarrow\mathcal{B}:=\{B_1\times B_2|(B_1\in\mathcal{B}_1)(B_2\in\mathcal{B}_2)\}\subseteq \{A_1\times A_2|(A_1\in\tau_1)(A_2\in\tau_2)\}\subseteq \tau_1\star\tau_2.$
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$(x,y)\in A\in\tau_1\star\tau_2$ olsun.
$\left.\begin{array}{r}A\in\tau_1\star\tau_2\Rightarrow (\exists \mathcal{A}_1\subseteq\tau_1)(\exists \mathcal{A}_2\subseteq\tau_2)(A=\cup_{(A_1\in\mathcal{A}_1)(A_2\in\mathcal{A}_2)}(A_1\times A_2)) \\ \\ (x,y)\in A\end{array}\right\} \Rightarrow$
$\left.\begin{array}{rr} \Rightarrow (\exists A_1\in\mathcal{A}_1\subseteq\tau_1)(\exists A_2\in\mathcal{A}_2\subseteq\tau_2)((x,y)\in A_1\times A_2) \\ \\ (\mathcal{B}_1, \,\ \tau_1 \text{ için baz})(\mathcal{B}_2, \,\ \tau_2 \text{ için baz}) \end{array}\right\}\Rightarrow $
$\Rightarrow (\exists \mathcal{A}_1^*\subseteq\mathcal{B}_1)(\exists \mathcal{A}_2^*\subseteq \mathcal{B}_2)(A_1=\cup\mathcal{A}_1^*)(A_2=\cup\mathcal{A}_2^*)((x,y)\in A_1\times A_2=(\cup\mathcal{A}_1^*)\times (\cup\mathcal{A}_2^*))$
$\Rightarrow (\exists B_1\in \mathcal{A}_1^*\subseteq\mathcal{B}_1)(\exists B_2\in \mathcal{A}_2^*\subseteq \mathcal{B}_2)((x,y)\in B_1\times B_2\subseteq A_1\times A_2\subseteq A)$
$\Rightarrow (B_1\times B_2\in \mathcal{B})((x,y)\in B_1\times B_2\subseteq A_1\times A_2\subseteq A).$