reserve C for Simple_closed_curve,
  A,A1,A2 for Subset of TOP-REAL 2,
  p,p1,p2,q ,q1,q2 for Point of TOP-REAL 2,
  n for Element of NAT;

theorem
  p in C implies p in Segment(p,W-min C,C) & W-min C in Segment(p,W-min C,C)
proof
A1: Segment(p,W-min C,C) = {p1: LE p,p1,C or p in C & p1=W-min C} by
JORDAN7:def 1;
  assume
A2: p in C;
  then LE p,p,C by JORDAN6:56;
  hence p in Segment(p,W-min C, C) by A1;
  thus thesis by A2,A1;
end;
