```markdown
## Proof

This is where we use the two technical assumptions.

- Then (4) implies that for all \( x \in \mathcal{D} \), \( \tilde{\nu}^T(Ax - b) \geq 0 \).
- But \( \tilde{x} \) satisfies \( \tilde{\nu}^T(A\tilde{x} - b) = 0 \), and since \( \tilde{x} \in \text{int} \, \mathcal{D} \), there are points in \( \mathcal{D} \) with \( \tilde{\nu}^T(Ax - b) < 0 \) unless \( A^T\tilde{\nu} = 0 \).
- This contradicts our assumption that \( \text{Rank} \, A = p \).

This means that we cannot have \( \mu = 0 \) and ends the proof.

*Source: A. d'Aspremont. Convex Optimization M2.*
```
