reserve x,y for set;
reserve s,r for Real;
reserve r1,r2 for Real;
reserve n for Nat;
reserve p,q,q1,q2 for Point of TOP-REAL 2;

theorem
  for P being Subset of TOP-REAL 2, q being Point of TOP-REAL 2
  st P is being_simple_closed_curve & q in P holds LE q,q,P
proof
  let P be Subset of TOP-REAL 2, q be Point of TOP-REAL 2;
  assume that
A1: P is being_simple_closed_curve and
A2: q in P;
A3: Upper_Arc(P) \/ Lower_Arc(P)=P by A1,Def9;
A4: Upper_Arc(P) is_an_arc_of W-min(P),E-max(P) by A1,Th50;
  now per cases by A2,A3,XBOOLE_0:def 3;
    case
   q in Upper_Arc(P);
      then LE q,q,Upper_Arc(P),W-min(P),E-max(P) by JORDAN5C:9;
      hence thesis;
    end;
    case
A5:   q in Lower_Arc(P) & not q in Upper_Arc(P);
      then
A6:   LE q,q,Lower_Arc(P),E-max(P),W-min(P) by JORDAN5C:9;
      q <> W-min P by A4,A5,TOPREAL1:1;
      hence thesis by A6;
    end;
  end;
  hence thesis;
end;
