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
  p in C & q in C implies LE p,q,C or LE q,p,C
proof
  assume that
A1: p in C and
A2: q in C;
A3: C = Lower_Arc C \/ Upper_Arc C by JORDAN6:50;
  per cases by A1,A2,A3,JORDAN7:1,XBOOLE_0:def 3;
  suppose
    p = q;
    hence thesis by A1,JORDAN6:56;
  end;
  suppose that
A4: p in Lower_Arc C & p <> W-min C & q in Lower_Arc C & q <> W-min C and
A5: p <> q;
    Lower_Arc C is_an_arc_of E-max C,W-min C by JORDAN6:50;
    then LE p,q,Lower_Arc C, E-max C,W-min C or LE q,p,Lower_Arc C, E-max C,
    W-min C by A4,A5,JORDAN5C:14;
    hence thesis by A4,JORDAN6:def 10;
  end;
  suppose
    p in Lower_Arc C & p <> W-min C & q in Upper_Arc C;
    hence thesis by JORDAN6:def 10;
  end;
  suppose
    p in Upper_Arc C & q in Lower_Arc C & q <> W-min C;
    hence thesis by JORDAN6:def 10;
  end;
  suppose that
A6: p in Upper_Arc C & q in Upper_Arc C and
A7: p <> q;
    Upper_Arc C is_an_arc_of W-min C, E-max C by JORDAN6:50;
    then LE p,q,Upper_Arc C,W-min C, E-max C or LE q,p,Upper_Arc C,W-min C,
    E-max C by A6,A7,JORDAN5C:14;
    hence thesis by A6,JORDAN6:def 10;
  end;
end;
