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Proof in Linear programming course

Please help me to find out how I can do this

(See attached file for full problem description)

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Let (see attachment)
It is clear that we can rewrite (attached) as (attached) , i.e. as a system of linear inequalities. (I've done this). Show that in fact we can rewrite (attached) as a system of (attached) linear inequalities.
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(See attached file for full problem description)

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Remark: there is a formal error in the demonstration of the problem, but the theorem which is to be proved is OK.

Notations:

Let a_i be the i-th row of ...

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