1. \(451_{6} - P_{7} = 305_{6}\)

2. \(P_{7} = 451_{6} - 305_{6}\)

3. \(P_{7} = 142_{6}\)

4. Convert \(142_{6} = 1 \cdot 6^{2} + 4 \cdot 6^{1} + 2 \cdot 6^{0}\)

5. \(= 36 + 24 + 2\)

6. \(= 62\)

7. Convert \(62_{10}\) to base 7:

\(\frac{62}{7} = 8 \text{ R } 6\)

\(\frac{8}{7} = 1 \text{ R } 1\)

\(\frac{1}{7} = 0 \text{ R } 1\)

8. Therefore, \(P_{7} = 116_{7}\).