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

theorem Th3:
  for P being compact non empty Subset of TOP-REAL 2,q st P is
  being_simple_closed_curve & q in P holds LE W-min(P),q,P
proof
  let P be compact non empty Subset of TOP-REAL 2,q;
  assume that
A1: P is being_simple_closed_curve and
A2: q in P;
A3: q in Upper_Arc(P) \/ Lower_Arc(P) by A1,A2,JORDAN6:50;
A4: Upper_Arc(P) is_an_arc_of W-min(P),E-max(P) by A1,JORDAN6:50;
A5: W-min(P) in Upper_Arc(P) by A1,Th1;
  per cases by A3,XBOOLE_0:def 3;
  suppose
A6: q in Upper_Arc(P);
    then LE W-min(P),q,Upper_Arc(P),W-min(P),E-max(P) by A4,JORDAN5C:10;
    hence thesis by A5,A6,JORDAN6:def 10;
  end;
  suppose
A7: q in Lower_Arc(P);
    per cases;
    suppose
      not q=W-min(P);
      hence thesis by A5,A7,JORDAN6:def 10;
    end;
    suppose
A8:   q=W-min(P);
      then LE W-min(P),q,Upper_Arc(P),W-min(P),E-max(P) by A5,JORDAN5C:9;
      hence thesis by A5,A8,JORDAN6:def 10;
    end;
  end;
end;
