= Solution
Suppose the <non-vertical supporting hyperplane of a value function> has slope $\lambda$. Every $x\in X$ then satisfies
$$
f(x)\geq\phi(h(x))\geq\phi(b)+\lambda^T(h(x)-b),
$$
so $L_b(x,\lambda)\geq\phi(b)$. Taking the infimum gives $\inf_XL_b\geq\phi(b)$. Conversely, feasible points approaching the infimum of $f$ have $L_b=f$, so $\inf_XL_b\leq\phi(b)$ even if no primal minimizer exists. Therefore
$$
\boxed{\inf_XL_b(x,\lambda)=\phi(b),}
$$
which is the <Strong Lagrangian property>.
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