# Dimension of a constructible set intersecting each orbit of a $G$-variety

In preparing a talk I'm having trouble with exercise 3 and 4 on page 25 of the following Lecture Notes of Crawley-Boevey (I only need the case $X=Y$ there):

$\text{3.}$ Let $X$ be a variety with a connected algebraic group $G$ acting on it. Let $X_s$ be the union of all orbits of dimension $s$ (or of dimension $\leq s$ if you prefer). Let $Z$ be a constructible subset intersecting each orbit, then for all $s$ we have $$\dim X_s-s\leq \dim Z$$ $\text{4.}$ Let $f:Z\to X$ be a morphism with inverse image of each orbit having dimension $\leq d$, then we have: $$\max_s\dim X_s-s\geq \dim Z-d$$

If someone could provide a hint, a reference, or a proof to either of the exercises, that would be great. In the notes there are not that many theorems that seem helpful to me: Maybe one could use the statement about fibre dimension of dominant morphisms or the orbit formula.

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