Hello,
I have to show: if I have two structures \(\displaystyle \mathcal{A}, \mathcal{B}\), and the Spoiler have a winning strategy in the game \(\displaystyle \mathfrak{G}_m(\mathcal{A},\mathcal{B})\) then there exists a \(\displaystyle m' \in \mathbb{N}\) for which the Spoiler have a \(\displaystyle \mathcal{A}\)-restricted or \(\displaystyle \mathcal{B}\)-restricted winning strategy in the game \(\displaystyle \mathfrak{G}_{m'}(\mathcal{A},\mathcal{B})\) (means he only can take elements from the structure A or only from B).
How to show this?
My idea is to show it by contradiction: the spoiler have a winning strategy in \(\displaystyle \mathfrak{G}_m(\mathcal{A},\mathcal{B})\) and there is NO such \(\displaystyle m' \in \mathbb{N}\) so that he has a \(\displaystyle \mathcal{A}\)-restricted or \(\displaystyle \mathcal{B}\) winning strategy in the game \(\displaystyle \mathfrak{G}_{m'}(\mathcal{A},\mathcal{B})\).
But how to come to the contradiction here?
Thx!
I have to show: if I have two structures \(\displaystyle \mathcal{A}, \mathcal{B}\), and the Spoiler have a winning strategy in the game \(\displaystyle \mathfrak{G}_m(\mathcal{A},\mathcal{B})\) then there exists a \(\displaystyle m' \in \mathbb{N}\) for which the Spoiler have a \(\displaystyle \mathcal{A}\)-restricted or \(\displaystyle \mathcal{B}\)-restricted winning strategy in the game \(\displaystyle \mathfrak{G}_{m'}(\mathcal{A},\mathcal{B})\) (means he only can take elements from the structure A or only from B).
How to show this?
My idea is to show it by contradiction: the spoiler have a winning strategy in \(\displaystyle \mathfrak{G}_m(\mathcal{A},\mathcal{B})\) and there is NO such \(\displaystyle m' \in \mathbb{N}\) so that he has a \(\displaystyle \mathcal{A}\)-restricted or \(\displaystyle \mathcal{B}\) winning strategy in the game \(\displaystyle \mathfrak{G}_{m'}(\mathcal{A},\mathcal{B})\).
But how to come to the contradiction here?
Thx!