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 Th5:
  LE q1, q2, A, p1, p2 implies q1 in Segment(A,p1,p2,q1,q2) & q2 in
  Segment(A,p1,p2,q1,q2)
proof
A1: Segment(A,p1,p2,q1,q2) = R_Segment(A,p1,p2,q1) /\ L_Segment(A,p1,p2,q2)
  by JORDAN6:def 5;
  assume
A2: LE q1, q2, A, p1, p2;
  L_Segment(A,p1,p2,q2) = {q : LE q,q2,A,p1,p2} by JORDAN6:def 3;
  then
A3: q1 in L_Segment(A,p1,p2,q2) by A2;
  q1 in A by A2,JORDAN5C:def 3;
  then q1 in R_Segment(A,p1,p2,q1) by Th4;
  hence q1 in Segment(A,p1,p2,q1,q2) by A1,A3,XBOOLE_0:def 4;
  R_Segment(A,p1,p2,q1) = {q : LE q1,q,A,p1,p2} by JORDAN6:def 4;
  then
A4: q2 in R_Segment(A,p1,p2,q1) by A2;
  q2 in A by A2,JORDAN5C:def 3;
  then q2 in L_Segment(A,p1,p2,q2) by Th3;
  hence thesis by A1,A4,XBOOLE_0:def 4;
end;
