I have a probability problem as follow:
$\mathbb{P}\big[\mathop{\arg\sup}_{x \in \bigcup_{i\in \{1,2\}} \Phi_{k,i} } \mathcal{f}(x )\geq y \big] = 1-\mathop{\prod}_{x \in \bigcup_{i\in \{1,2\}} \Phi_{k,i} } \mathbb{P}\big[ \mathcal{f}(x )\leq y \big]$
$\Phi_k$ is a 2D Poisson Point process, so $x$ is a point. $f : \mathbb{R}^2\longrightarrow \mathbb{R}$
This might be a simple question but I need some hint. How can I expand the product?