let P be Subset of (TOP-REAL 2); :: thesis: for q being Point of (TOP-REAL 2) st P is being_simple_closed_curve & q in P holds
LE q,q,P
let q be Point of (TOP-REAL 2); :: thesis: ( P is being_simple_closed_curve & q in P implies LE q,q,P )
assume that
A1: P is being_simple_closed_curve and
A2: q in P ; :: thesis: LE q,q,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 :: thesis: ( ( q in Upper_Arc P & LE q,q,P ) or ( q in Lower_Arc P & not q in Upper_Arc P & LE q,q,P ) )

per cases ( q in Upper_Arc P or ( q in Lower_Arc P & not q in Upper_Arc P ) ) by A2, A3, XBOOLE_0:def 3;

case q in Upper_Arc P ; :: thesis: LE q,q,P

then LE q,q, Upper_Arc P, W-min P, E-max P by JORDAN5C:9;
hence LE q,q,P ; :: thesis: verum

end;

case A5: ( q in Lower_Arc P & not q in Upper_Arc P ) ; :: thesis: LE q,q,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 LE q,q,P by A6; :: thesis: verum

end;

end;

end;

hence LE q,q,P ; :: thesis: verum