Every polyhedron is convex set(proof explanation) 
Everything is clear, except one thing, that i didn't get:
How do the author of proof make the following jump

How exactly he got last inequality?

It follows by linearity and applying the inequalities you're given. 
$\mathbf{a}'(\lambda \mathbf{x} + (1- \lambda)\mathbf{y}) = \lambda\mathbf{a}' \mathbf{x} + (1- \lambda)\mathbf{a}'\mathbf{y} \geq \lambda b + (1-\lambda)b = \lambda b + b - \lambda b = b$
