  In a recent paper, the complete (non-linear) Kaluza-Klein Ansatz for the
consistent embedding of certain scalar plus gravity subsectors of gauged
maximal supergravity in D=4, 5 and 7 was presented, in terms of sphere
reductions from D=11 or type IIB supergravity. The scalar fields included in
the truncations were the diagonal fields in the SL(N,R)/SO(N) scalar
submanifolds of the full scalar sectors of the corresponding maximal
supergravities, with N=8, 6 and 5. The embeddings were used for obtaining an
interpretation of extremal D=4, 5 or 7 AdS domain walls in terms of distributed
M-branes or D-branes in the higher dimensions. Although strong supporting
evidence for the correctness of the embedding Ansatze was presented, a full
proof of the consistency was not given. Here, we complete the proof, by showing
explicitly that the full set of higher-dimensional equations of motion are
satisfied if and only if the lower-dimensional fields satisfy the relevant
scalar plus gravity equations.
