1. Problem: Simplify the expression $ \frac{x - z}{y - z} \cdot \left( \frac{y - z}{x - z} : \frac{y - z}{x - z} \right) $. 2. Recall that division of fractions means multiplying by the reciprocal: $ a : b = a \cdot \frac{1}{b} $. 3. Simplify the inner division: $$\frac{y - z}{x - z} : \frac{y - z}{x - z} = \frac{y - z}{x - z} \cdot \frac{x - z}{y - z} = 1$$ 4. Substitute back: $$\frac{x - z}{y - z} \cdot 1 = \frac{x - z}{y - z}$$ 5. Final simplified form is: $$\boxed{\frac{x - z}{y - z}}$$ This means the original expression simplifies to $ \frac{x - z}{y - z} $ because the division inside the parentheses equals 1.