Here are the paraphrases for "The cat chased the mouse" without using the article "the". I have replaced "the" with "a":

A feline hunted a rodent.
A mouse ran away from a cat.
A cat followed a mouse closely.
A rodent was chased by a feline.
A cat ran after a mouse.
A mouse tried to escape a cat.
A feline pursued a rodent.
A mouse was followed by a cat.
A cat sprinted towards a mouse.
A rodent ran from a feline.
A cat stalked a mouse.
A mouse avoided a cat.
A feline raced after a rodent.
A mouse was hunted by a cat.
A cat dashed towards a mouse.
A mouse fled from a cat.
A feline tracked a rodent.
A mouse was stalked by a cat.
A cat lunged at a mouse.
A rodent escaped a feline.
A cat went after a mouse.
A mouse dodged a cat.
A feline ran towards a rodent.
A mouse was sprinted at by a cat.
A cat pounced on a mouse.
A rodent eluded a feline.
A cat chased down a mouse.
A mouse hid from a cat.
A feline caught up with a rodent.
A mouse was pounced on by a cat.

Let me know if you need further rephrases.
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