To simplify the expression $\frac{2}{5} \text{ of } \left(\left(\frac{3}{4} \div 0.25\right) + \frac{1}{2}\right) - \frac{1}{3}$, we follow the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Division and Multiplication (from left to right), Addition and Subtraction (from left to right).

Step-by-Step Simplification

1. Convert decimal to fraction: The decimal $0.25$ is equal to $\frac{25}{100}$, which simplifies to $\frac{1}{4}$.

   $0.25 = \frac{1}{4}$

2. Solve the division inside the innermost parentheses: Substitute the fraction form of $0.25$. Division by a fraction is multiplication by its reciprocal.

   $ \frac{3}{4} \div 0.25 = \frac{3}{4} \div \frac{1}{4} = \frac{3}{4} \times \frac{4}{1} = \frac{12}{4} = 3 $

3. Solve the addition inside the main parentheses: Now add $\frac{1}{2}$ to the result from the previous step.

   $ 3 + \frac{1}{2} = \frac{6}{2} + \frac{1}{2} = \frac{7}{2} $

4. Perform the multiplication ('of'): Multiply $\frac{2}{5}$ by the result from the parentheses ($\frac{7}{2}$).

   $ \frac{2}{5} \text{ of } \frac{7}{2} = \frac{2}{5} \times \frac{7}{2} = \frac{2 \times 7}{5 \times 2} = \frac{14}{10} = \frac{7}{5} $

5. Perform the final subtraction: Subtract $\frac{1}{3}$ from the result obtained.

   $ \frac{7}{5} - \frac{1}{3} $

   To subtract, find a common denominator, which is $15$. Convert the fractions:

   $ \frac{7 \times 3}{5 \times 3} - \frac{1 \times 5}{3 \times 5} = \frac{21}{15} - \frac{5}{15} = \frac{21 - 5}{15} = \frac{16}{15} $

The simplified expression equals $\frac{16}{15}$.