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       The horse exerts force on the ground.  It pushes the ground backwards.  Then the ground exerts force on the horse in the forward direction.  So it moves forward relative to the ground.  The ground being massive, moves very little backwards.

     The horse is attached to the cart.  The attachments and hence the cart is pulled by the horse in the forward direction.  The cart moves forward due to the force exerted by the horse on the cart.

   The cart exerts equal and opposite force on the horse.  So  the horse does not move with respect to the cart.  Both move together as they exert forces equal and opposite.

   The force that is causing both cart and horse is the reaction force from the ground on the horse.

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