Proof:

This can be proved by showing any point on the preimage is translated a distance twice that of the distance between the lines:

Suppose the distance between the parallel lines is d, and consider a point P on the preimage.  If P is a distance x from l₁, then its image P' upon reflection in l₁ is a distance x on the other side of l₁, and is therefore a distance d – x from l₂.  When P' is reflected in l₂, then its image Q is a distance d – x on the other side of l₂.  The distance from P to Q is equal to the distance from P to l₁ plus the distance from l₁ to P' plus the distance from P' to l₂ plus the distance from l₂ to Q:  PQ = x + x + d – x + d – x = 2d.

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