Ok looking at your question again I can see that it's probably the second property that I wrote that is the one you have gotten mixed up. It's the curl of a grad that is zero (a zero vector) not the way you put it, grad of a curl is not neccessarily zero
that the curl of the grad is zero is essentially trivial. it follows from the trivial fact that the integral of GRAD is zero around a closed curve. this is because that integral is evaluated by subtracting the values of some function at the two endpoints, which are equal.
that in turn is true by the FTC, which holds by the trivial fact that the derivative of the area function is the height functon, which holds trivially because the area of a rectangle is the base times the height.
on the other hand, sticking all these trivialities together, maybe we have a pathologically amazing theorem!
more interesting is the investigation of forms with zero curl which are not gradients, like dtheta.
One other thing I should have noticed about the original post. "Grad of Curl" isn't even defined. Grad is normally applied to a scaler field, giving a vector result. So "div of curl" makes sense but "grad of curl" doesn't. So my guess is that the OP is getting confused between grad and div.