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 Th4:
  q in A implies q in R_Segment(A,p1,p2,q)
proof
  assume q in A;
  then
A1: LE q,q,A,p1,p2 by JORDAN5C:9;
  R_Segment(A,p1,p2,q) = {q1 : LE q,q1,A,p1,p2} by JORDAN6:def 4;
  hence thesis by A1;
end;
