Look at a simplified example. Use bit strings of lenth 3, P1 the strings that agree on the last 2 bits and P2 the stings that agree on the last bit.

There are 8 strings of length 3: 000, 001, 010, 011, 100, 101, 110, and 111.

P1 contains 8/2= 4 sets: {000, 100}, {010, 110}, {001, 101}, and {011, 111}.

P2 contains 8/4= 2 sets: {000, 100, 010, 110} and {001, 101, 011, 111}.

Each member of P1 is a subset of P2.