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Let assume sets X, Y, Z are not finite.
Since if we take X, Y, and Z are finite then result is trivial.
Now, we have |X| = |Y| and |Y| = |Z|.
Then we can define a bijection from X to Y says f
f : X → Y is bijection
Also there is a bijection g from Y onto Z
g : Y → Z is bijection
The composition of two bijective functions is bijective.
This shows that g○f is bijection from X to Z.
Since rang of f and domain of g is the same set, so the composition mapping is well defined.
Let h = g○f
Thus h is a bijection from set X onto set Z.
h : X → Z is bijection
|X| = |Z|