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The main result of this paper is the construction of a strictly positive extension of any no-arbitrage price system defined on an incomplete market of any dimension and for any (finite) number of states of the world. This is proved by using the component functionals of the positive basis of the 2-dimensional lattice-subspace, which is spanned by the riskless asset and any of the state-discriminating payoffs, which are generically existent in the span of the primitive, non-redundant assets of the incomplete market, whose time-period 1 payoffs are x1, x2,···, xn, respectively. This result is also valid in the case where the market is complete. Both of these results lead to a new statement of both of the Fundamental Theorems of Asset Pricing in the finite-state case.

7BThe 2-Dimensional Lattice-Subspaces in Finite-State Finance