Statement

Lemma

For a linear programme given by $A, b,$ and $c$ if we take the dual linear programme and then the dual linear programme of that we get back to the original linear programme.

Proof

Just following the definition the dual linear programme is defined by $-A^T$, $-c$ and $-b$. Then the dual linear programme of that is given by $-(-A^T)^T$, $-(-b)$ and $-(-c)$ which by laws of linear algebra give us $A$, $b$ and $c$ back.