reserve p,p1,p2,p3,q for Point of TOP-REAL 2;

theorem Th14:
  for P being compact non empty Subset of TOP-REAL 2, q1,q2,q3
  being Point of TOP-REAL 2 st P is being_simple_closed_curve & LE q1,q2,P & LE
q2,q3,P & q1<>W-min P & q2<>q3 holds Segment(q1,q2,P) misses Segment(q3,W-min P
  ,P)
proof
  let P be compact non empty Subset of TOP-REAL 2, q1,q2,q3 be Point of
  TOP-REAL 2;
  assume that
A1: P is being_simple_closed_curve and
A2: LE q1,q2,P and
A3: LE q2,q3,P and
A4: q1<>W-min P and
A5: q2<>q3;
  set x = the Element of Segment(q1,q2,P)/\ Segment(q3,W-min P,P);
  assume
A6: Segment(q1,q2,P)/\ Segment(q3,W-min P,P)<>{};
  then
A7: x in Segment(q1,q2,P) by XBOOLE_0:def 4;
  x in Segment(q3,W-min P,P) by A6,XBOOLE_0:def 4;
  then x in {p1: LE q3,p1,P or q3 in P & p1=W-min P} by Def1;
  then
A8: ex p1 st p1=x &( LE q3,p1,P or q3 in P & p1=W-min P);
  q2 <> W-min P by A1,A2,A4,Th2;
  then x in {p: LE q1,p,P & LE p,q2,P} by A7,Def1;
  then ex p3 st p3 = x & LE q1,p3,P & LE p3,q2,P;
  then LE q3,q2,P by A1,A4,A8,Th2,JORDAN6:58;
  hence contradiction by A1,A3,A5,JORDAN6:57;
end;
