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GoBoard Theorem
, 1999
"... this paper. For simplicity, we adopt the following convention: p, p 1 , p 2 , q 1 , q 2 denote points of E ..."
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this paper. For simplicity, we adopt the following convention: p, p 1 , p 2 , q 1 , q 2 denote points of E
On the rectangular finite sequences of the points of the plane
 Journal of Formalized Mathematics
, 1997
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Decomposing a GoBoard into Cells MML Identifier:GOBOARD5. Yatsuka Nakamura
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Summary. This is the continuation of the proof of the Jordan Theorem according to
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Some Properties of Cells on Go Board MML Identifier:GOBRD13.
"... non empty set, and f denotes a finite sequence of elements of D. Let E be a non empty set, let S be a non empty set of finite sequences of the carrier ofE 2 T, let F be a function from E into S, and let e be an element of E. Then F(e) is a finite sequence of elements ofE 2 T. Let F be a function. Th ..."
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non empty set, and f denotes a finite sequence of elements of D. Let E be a non empty set, let S be a non empty set of finite sequences of the carrier ofE 2 T, let F be a function from E into S, and let e be an element of E. Then F(e) is a finite sequence of elements ofE 2 T. Let F be a function. The functor ValuesF yields a set and is defined by: (Def. 1) ValuesF = � (rng κ F(κ)). One can prove the following proposition (1) For every finite sequence M of elements of D ∗ holds M(i) is a finite sequence of elements of D. Let D be a set. Observe that every finite sequence of elements of D ∗ is finite sequence yielding. One can check that every function which is finite sequence yielding is also function yielding. One can prove the following proposition (3) 1 For every finite sequence M of elements of D ∗ holds ValuesM = � {rng f; f ranges over elements of D ∗ : f ∈ rngM}. Let D be a non empty set and let M be a finite sequence of elements of D ∗. One can check that ValuesM is finite. One can prove the following propositions: (4) For every matrix M over D such that i ∈ domM and M(i) = f holds len f = widthM. (5) For every matrix M over D such that i ∈ domM and M(i) = f and j ∈ dom f holds 〈i, j 〉 ∈ the indices of M. (6) For every matrix M over D such that 〈i, j 〉 ∈ the indices of M and M(i) = f holds len f = widthM and j ∈ dom f. (7) For every matrix M over D holds ValuesM = {M ◦(i, j) : 〈i, j 〉 ∈ the indices of M}. (8) For every non empty set D and for every matrix M over D holds cardValuesM ≤ lenM · widthM. 1 The proposition (2) has been removed.