**Puzzle 173**
If we let \\(y = f(x')\\) then \\(f(x) \leq y\\) sets up the following inequalities by definition of a monotone function :
$$x \leq x' \;\text{and}\; f(x) \leq f(x')$$
By definition of a feasibility relation the above inequality implies :
$$\Phi(x,f(x)) \;\text{implies}\; \Phi(x',f(x'))$$
So...
$$\text{If} \; \Phi(x,f(x)) = \text{true and} \; f(x) \leq y, \text{then} \; \Phi(x,y) = \text{true}$$
**Puzzle 174**
Similarly,
$$\text{If} \; \Psi(g(y),y) = \text{true and} \; x \leq g(y), \text{then} \; \Psi(x,y) = \text{true}$$