It's not true. Here is a counter-example:

Let X = {a,b,c,d}, Y = {b,c,e,f}, Z = {c,d,f,g}

Then Z - (Y - X) = {c,d,f,g} - {e,f} = {c,d,g} whereas:

X U (Z - Y) = {a,b,c,d} U {d,g} = {a,b,c,d,g}, and these two sets are clearly unequal.

It is probably a typo, perhaps the author intended:

Z - (Y - X) is a subset of X U (Z - Y), or:

Z - (Y - X) = (X ∩ Y ∩ Z) U (Z - Y), it's hard to say.