Convex Real Function is Pointwise Supremum of Affine Functions

I'm read the proof of the Convex Real Function is Pointwise Supremum of Affine Functions

https://proofwiki.org/wiki/Convex_Real_Function_is_Pointwise_Supremum_of_Affine_Functions

But I dont understand the afirmation:

for all $x,y\in\mathbb{R}$with $x\neq y$we have:

$f(y)≥f(x)+c_x(y−x)$ , with $c_x:=\sup\{\frac{f(y)-f(x)}{y-x}:y<x \} $

Because for $y<x$ ,then:

$\frac{f(y)-f(x)}{y-x}\leq c_x\implies f(y)\leq f(x)+c_x(y-x)$