No, Newton's law is not violated, because frictional forces are acting.
Work Step by Step
If you could magically turn off all friction acting on the cart and its wheels, it would continue in motion even after you stopped pushing. But in the real world, when surfaces rub together, there is always some friction, so the cart slows. This situation doesn’t violate Newton's first law, because there is an external force. This is discussed on page 26.