  We give a presentation of the torus-equivariant quantum $K$-theory ring of
flag manifolds of type $A$, as a quotient of a polynomial ring by an explicit
ideal. This is the torus-equivariant version of our previous result, which
gives a presentation of the non-equivariant quantum $K$-theory ring of flag
manifolds of type $A$. However, the method of proof for the torus-equivariant
one is completely different from that for the non-equivariant one; our proof is
based on the result in the $Q = 0$ limit, and uses Nakayama-type arguments to
upgrade it to the quantum situation. Also, in contrast to the non-equivariant
case in which we used the Chevalley formula, we make use of the inverse
Chevalley formula for the torus-equivariant $K$-group of semi-infinite flag
manifolds to obtain a relation which yields our presentation.
