Upper and Lower Sequence of a Cage1
نویسنده
چکیده
(1) For every non empty subset X of E2 T and for every compact subset Y of E2 T such that X ⊆Y holds N-bound(X)≤ N-bound(Y ). (2) For every non empty subset X of E2 T and for every compact subset Y of E2 T such that X ⊆Y holds E-bound(X)≤ E-bound(Y ). (3) For every non empty subset X of E2 T and for every compact subset Y of E2 T such that X ⊆Y holds S-bound(X)≥ S-bound(Y ). (4) For every non empty subset X of E2 T and for every compact subset Y of E2 T such that X ⊆Y holds W-bound(X)≥W-bound(Y ). (5) Let f , g be finite sequences of elements of E2 T. Suppose f is in the area of g. Let p be an element of E2 T. If p ∈ rng f , then f −: p is in the area of g. (6) Let f , g be finite sequences of elements of E2 T. Suppose f is in the area of g. Let p be an element of E2 T. If p ∈ rng f , then f :− p is in the area of g. (7) For every non empty finite sequence f of elements of E2 T and for every point p of E2 T such that p ∈ L̃( f ) holds p, f 6= / 0. (8) Let f be a non empty finite sequence of elements of E2 T and p be a point of E2 T. If p∈ L̃( f ) and len f , p≥ 2, then f (1) ∈ L̃( f , p). (9) Let f be a non empty finite sequence of elements of E2 T. Suppose f is a special sequence. Let p be a point of E2 T. If p ∈ L̃( f ), then f (1) / ∈ L̃(mid( f , Index(p, f )+1, len f )). (10) For all natural numbers i, j, m, n such that i+ j = m+n and i≤ m and j ≤ n holds i = m. (11) Let f be a non empty finite sequence of elements of E2 T. Suppose f is a special sequence. Let p be a point of E2 T. If p ∈ L̃( f ) and f (1) ∈ L̃( p, f ), then f (1) = p. 1This work has been partially supported by CALCULEMUS grant HPRN-CT-2000-00102.
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