**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,

$$\text{If} \; \Psi(g(y),y) = \text{true and} \; x \leq g(y), \text{then} \; \Psi(x,y) = \text{true}$$