Why does $c^Tx \le d$ on an affine $D = \{x : Ax = b\}$ set imply that $c^Tx = d$?
I tried to proceed with a proof by contradiction assuming $c^Tx \not = d$ i.e. $c^Tx < d$. However, I don't see here any contradiction. Intuitively it is the intersection of a half-space with a hyperplane however, I am struggling to understand the problem.
It most certainly does not mean that. Take $c=\vec 0$ and $d=1$ for a counterexample.