If i want to trade in my car to a dealership, do i have to show proof that i brought it?

I brought my car which is a holden barina spark, i brought it brand new. My car is worth $14,000 brand new. But because holden was having a sale on that weekend i managed to get the car brand new for $12,500.00. I have had my car for 4 months only and it has done only 7000 kms. I was just wondering because if i tell the dealership i only paid $12,500 i wont get much so i wanna know if i can just lie or do they need proof stating how much i paid?