The name is absent

Proof: The first part can be derived by induction. The second part follows from
the same line of the proof of Theorem 2.2.                                    □

We point out that all theorems above will be constructively demonstrated by
the algorithm to be presented in the next section. Let us give some examples.
Example 1. We are given

P = { x ∈ R² I ajx ≤ bi, i = 1, 2, 3 }

where α₁ = (3,2)^τ, α₂ = (1, — l)^τ and α₃ = (—3, — l)^τ, b₁ = 1, b₂ = — 1 and
b₃ = 1. This example is shown in Figure 1 where there are three completely labeled
simplices. One of them is of type II. The other two are of type I.

Example 2. We are given

P = { X ∈ R² I ajx ≤ bi, i = 1,2, 3 }

where α₁ = (2, — l)^τ, α₂ = (3, l)^τ and α₃ = (—3,0)^τ, b₁ = 1, b₂ = 2 and b₃ = —1.
This example is illustrated in Figure 2 where there is a unique completely labeled
simplex of type II.

3 The algorithm

In this section we shall discuss how to operate the algorithm to End a completely
labeled simplex within a finite number of steps. In the rest of the section we assume
that the simplex P associated with matrix A is given such that

a. β(n₊ιp ≤ 0 for j = 1, ∙ ∙ ∙, n∙,

b. ац > 0 for i = 1, ∙ ∙ ∙, n;

c. Cii_j ≤ 0 and ∖ai_j I < ац for i ψ j, i, j = 1, ∙ ∙ ∙, n.

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