We shall discuss the difficulty in solving and numerically integrating differential-algebraic equation systems of the dx/dt = f(x,u), g(x) = 0 where x is in R^n and u is in R^m and m <= n. In this context we shall introduce a horizontal lift and its exponentiation toward construction of a solution. Especially, the solution and behavior of the algebraic variable is of interest. Cases where u can be rough (belong to fractional Hoelder space) are of interest. A numerical approximation that can produce useful results in computer simulations will be discussed.