Chains on a Grating in Euclidean Space1
نویسنده
چکیده
(1) For all real numbers x, y such that x < y there exists a real number z such that x < z and z < y. (2) For all real numbers x, y there exists a real number z such that x < z and y < z. The scheme FrSet 1 2 deals with a non empty set A , a non empty set B, a binary functor F yielding an element of A , and a binary predicate P , and states that: {F (x,y);x ranges over elements of B,y ranges over elements of B : P [x,y]} ⊆ A for all values of the parameters. Let B be a set and let A be a subset of B. Then 2A is a subset of 2B. Let X be a set. A subset of X is an element of 2X . Let d be a real natural number. Let us observe that d is zero if and only if: (Def. 1) d 6> 0. Let d be a natural number. Let us observe that d is zero if and only if: (Def. 2) d 6≥ 1. Let us observe that there exists a natural number which is non zero. In the sequel d is a non zero natural number. Let us consider d. Note that Segd is non empty. In the sequel i, i0 denote elements of Segd. Let us consider X . Let us observe that X is trivial if and only if: (Def. 3) For all x, y such that x ∈ X and y ∈ X holds x = y. We now state the proposition 1This work has been partially supported by CALCULEMUS grant HPRN-CT-2000-00102 and TYPES grant IST-1999-29001.
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