Preparing the Internal Approximations of Simple Closed Curves

The articles [18], [5], [20], [11], [1], [16], [2], [21], [4], [3], [12], [17], [7], [8], [9], [10], [13], [14], [15], [6], and [19] provide the terminology and notation for this paper. In this paper j, k, n are natural numbers and C is a subset of E T satisfying conditions of simple closed curve. Let us consider C. The functor ApproxIndexC yielding a natural number is defined by: (Def. 1) ApproxIndexC is sufficiently large for C and for every j such that j is sufficiently large for C holds j ­ ApproxIndexC. Next we state the proposition (1) ApproxIndexC ­ 1. Let us consider C. The functor Y-InitStartC yields a natural number and is defined as follows: (Def. 2) Y-InitStartC < widthGauge(C,ApproxIndexC) and cell(Gauge(C, ApproxIndexC),X-SpanStart(C,ApproxIndexC) − 1,Y-InitStartC) ⊆ BDDC and for every j such that j < widthGauge(C,ApproxIndexC) and cell(Gauge(C,ApproxIndexC),X-SpanStart(C,ApproxIndexC)− 1, j) ⊆ BDDC holds j ­ Y-InitStartC.