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
  LE q1, q2, A, p1, p2 implies Segment(A,p1,p2,q1,q2) is non empty
proof
A1: Segment(A,p1,p2,q1,q2)={q:LE q1,q,A,p1,p2 & LE q,q2,A,p1,p2} by JORDAN6:26;
  assume
A2: LE q1, q2, A, p1, p2;
  then q2 in A by JORDAN5C:def 3;
  then LE q2,q2,A,p1,p2 by JORDAN5C:9;
  then q2 in Segment(A,p1,p2,q1,q2) by A2,A1;
  hence thesis;
end;
