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The so-called minimax theorem means that if X and Y are two sets, and f and g are two real-valued functions defined on X×Y, then under some conditions the following inequality holds: $in{f}_{y\in Y}su{p}_{x\in X}f(x,y)\le su{p}_{x\in X}in{f}_{y\in Y}g(x,y)$. We will extend the two functions version of minimax theorems without the usual condition: f ≤ g. We replace it by a milder condition: $su{p}_{x\in X}f(x,y)\le su{p}_{x\in X}g(x,y)$, ∀y ∈ Y. However, we require some restrictions; such as, the functions f and g are jointly upward, and their upper sets are connected. On the other hand, by using some properties...