  We prove the long-standing conjecture on the coset construction of the
minimal series principal $W$-algebras of $ADE$ types in full generality. We do
this by first establishing Feigin's conjecture on the coset realization of the
universal principal $W$-algebras, which are not necessarily simple. As
consequences, the unitarity of the "discrete series" of principal $W$-algebras
is established, a second coset realization of rational and unitary $W$-algebras
of type $A$ and $D$ are given and the rationality of Kazama-Suzuki coset vertex
superalgebras is derived.
