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Check the two claims made in Proposition 1.53.
Proposition 1.53. For any preorder \( (P, \le_P ) \), the identity function is monotone.
If \( (Q, \le_Q ) \) and \( (R, \le_R) \) are preorders and \( f : P \rightarrow Q \) and \( g : Q \rightarrow R \) are monotone, then \( ( f .g) : P \rightarrow R \) is also monotone.