Upper and Lower Sequence on the Cage, Upper and Lower Arcs 1
Robert Milewski · 2002
The articles [25], [30], [2], [4], [3], [29], [5], [14], [27], [20], [24], [13], [1], [23], [10], [11], [8], [28], [16], [12], [21], [26], [7], [18], [19], [6], [22], [9], [15], and [17] provide the notation and terminology for this paper. In this paper n is a natural number. The following propositions are true: (1) Let G be a Go-board and i1, i2, j1, j2 be natural numbers. Suppose 1 ¬ j1 and j1 ¬ width G and 1 ¬ j2 and j2 ¬ width G and 1 ¬ i1 and i1 < i2 and i2 ¬ len G. Then (G ◦ (i1, j1))1 < (G ◦ (i2, j2))1. (2) Let G be a Go-board and i1, i2, j1, j2 be natural numbers. Suppose 1 ¬ i1 and i1 ¬ len G and 1 ¬ i2 and i2 ¬ len G and 1 ¬ j1 and j1 < j2 and j2 ¬ width G. Then (G ◦ (i1, j1))2 < (G ◦ (i2, j2))2. Let f be a non empty finite sequence and let g be a finite sequence. One can verify that f aa g is non empty. The following propositions are true: (3) Let C be a compact connected non vertical non horizontal subset of E2 T and n be a natural number. Then L(Cage(C, n)−:E-max L(Cage(C, n)))∩ L(Cage(C, n) :− E-max L(Cage(C, n))) = {N-min L(Cage(C, n)), E-max L(Cage(C, n))}. (4) For every compact connected non vertical non horizontal subset C of E2 T holds UpperSeq(C, n) = ((Cage(C, n))E-max e L(Cage(C,n)) a ) :− W-min L(Cage(C, n)).