  Using the covariant N=2 harmonic supergraph techniques we calculate the
one-loop low-energy effective action of N=4 super-Yang-Mills theory in Coulomb
branch with gauge group SU(2) spontaneously broken down to U(1). The full
dependence of the low-energy effective action on both the hypermultiplet and
gauge fields is determined. The direct quantum calculation confirms the
correctness of the exact N=4 SYM low-energy effective action derived in
hep-th/0111062 on the purely algebraic ground by invoking a hidden N=2
supersymmetry which completes the manifest N=2 one to N=4. Our results provide
an exhaustive solution to the problem of finding out the exact completely N=4
supersymmetric low-energy effective action for the theory under consideration.
