Two zero-sum games have saddle points. Does the sum of these games have a saddle point?

Question

Consider two zero-sum games. In each game, the set of strategies for each player is the $[0,1]$ segment. Let $x$ and $y$ be generic strategies of players 1 and 2. In the first game, the utility function of Player 1 is the function $U[x,y]$ which is continuous in $x$ and $y$, and the utility of Player 2 is $-U[x,y]$. In the second game, the function $U[x,y]$ is replaced by the function $F[x,y]$ which is also continuous in $x$ and $y$. Suppose that each game has a saddle point. If we consider a game where the utility function of Player is $U[x,y]+F[x,y]$, does this game also have a saddle point?