Sum of concave and strictly concave functions How can I prove, for the general case, that the sum of a concave and a strictly concave function, yields a strictly concave function?

$\color{blue}f$ concave, $\color{olive}g$ strictly concave, $t \in ]0,1[$

\begin{align} \color{blue}f(tx+(1-t)y)\,+\,\color{olive}g(tx+(1-t)y) &\color{blue}\geq t\color{blue}f(x)+(1-t)\color{blue}f(y)\,+\,\color{olive}g(tx+(1-t)y) \\\ &\color{olive}> t\color{blue}f(x)+(1-t)\color{blue}f(y)\,+\, t\color{olive}g(x)+(1-t)\color{olive}g(y) \\\ &= t(\color{blue}f(x)+\color{olive}g(x))\,+\,(1-t)(\color{blue}f(y)+\color{olive}g(y)) \end{align}

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