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Prove: The cross ratio of four points z_{1}, z_{2}, z_{3}, z_{4} is invariant under a bilinear transformation
i.e. if w_{1}, w_{2}, w_{3}, w_{4} are, respectively, the images of z_{1}, z_{2}, z_{3}, z_{4} under a bilinear transformation,
then the cross ratio of w_{1}, w_{2}, w_{3}, w_{4} must equal the cross ratio of z_{1}, z_{2}, z_{3}, z_{4}. In other words,
prove

Proof. Starting with

we first observe that

Consequently,

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