Adjacency of basic feasible solution and hyperplanes

Question

Problem:

Linear Programming : Adjacency of Basic Feasible Solution and Hyperplanes

Can anyone finish up this proof by continuing my preliminery work?

Assume A ∈ R^mxn b€R^m, with rank (A) = m are given. Two different basic feasible solutions u, v of Ax=b, x≥0 are said to be adjacent if they correspond to two bases that have exactly m-1 elements in common.

Suppose u, v are two different basic feasible solutions of Ax=b,x≥0. Prove that u and v are adjacent if and only if there exists a supporting hyperplane H of P:={x:Ax=b,x≥0.} such that H∩P = conv{u,v}

Let Z = λu+(1-λ)v,satisfying AZ = b

We can rewrite as a_i^T Z = bi

                          → Σ a_j^T Z = Σ_jbj

                          → Σ a [λu+(1-λ)v]= Σb_j

                          → λΣ a_j^T u +(1-λ)Σ a_j^T v = Σ b_j

                          → Σ a_jv+λΣa_j(u-v) = Σ b_j

Since Σ a_j v = Σbj        (u,v are bfs of AX=b,x≥0)

         Σa_j(u-v)=0         Thus u and v are in the same supporting hyperplane H.