let C be Simple_closed_curve; :: thesis: for p being Point of (TOP-REAL 2) holds L_Segment (Lower_Arc C),(E-max C),(W-min C),p = Segment (Lower_Arc C),(E-max C),(W-min C),(E-max C),p
let p be Point of (TOP-REAL 2); :: thesis: L_Segment (Lower_Arc C),(E-max C),(W-min C),p = Segment (Lower_Arc C),(E-max C),(W-min C),(E-max C),p
Lower_Arc C is_an_arc_of E-max C, W-min C by JORDAN6:65;
then R_Segment (Lower_Arc C),(E-max C),(W-min C),(E-max C) = Lower_Arc C by JORDAN6:27;
hence Segment (Lower_Arc C),(E-max C),(W-min C),(E-max C),p = (Lower_Arc C) /\ (L_Segment (Lower_Arc C),(E-max C),(W-min C),p) by JORDAN6:def 5
.= L_Segment (Lower_Arc C),(E-max C),(W-min C),p by JORDAN6:20, XBOOLE_1:28 ;
:: thesis: verum