On the Dividing Function of the Simple Closed Curve into Segments

In this paper p, p1, q are points of E 2 T . The following three propositions are true: (1) Let P be a compact non empty subset of E T . Suppose P is a simple closed curve. Then W-minP ∈ LowerArcP and E-maxP ∈ LowerArcP and W-minP ∈ UpperArcP and E-maxP ∈ UpperArcP. (2) For every compact non empty subset P of E T and for every q such that P is a simple closed curve and LE(q,W-minP, P ) holds q = W-minP. (3) For every compact non empty subset P of E T and for every q such that P is a simple closed curve and q ∈ P holds LE(W-minP, q, P ).