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
  Lower_Arc C <> Upper_Arc C
proof
  assume Lower_Arc C = Upper_Arc C;
  then
A1: Lower_Arc C =(C\Lower_Arc C) \/ {W-min C, E-max C} by JORDAN6:51;
  Lower_Arc C is_an_arc_of W-min C, E-max C by JORDAN6:50;
  then
A2: ex p3 being Point of TOP-REAL 2 st p3 in Lower_Arc C & p3<>W-min C & p3
  <>E-max C by JORDAN6:42;
  Lower_Arc C misses C\Lower_Arc C by XBOOLE_1:79;
  then Lower_Arc C c= {W-min C, E-max C} by A1,XBOOLE_1:73;
  hence contradiction by A2,TARSKI:def 2;
end;
