let C be Simple_closed_curve; :: thesis: Lower_Middle_Point C in Lower_Arc C

set L = Vertical_Line (((W-bound C) + (E-bound C)) / 2);

set p = First_Point ((Lower_Arc C),(W-min C),(E-max C),(Vertical_Line (((W-bound C) + (E-bound C)) / 2)));

A1: Lower_Arc C meets Vertical_Line (((W-bound C) + (E-bound C)) / 2) by Th62;

Vertical_Line (((W-bound C) + (E-bound C)) / 2) is closed by Th6;

then A2: (Lower_Arc C) /\ (Vertical_Line (((W-bound C) + (E-bound C)) / 2)) is closed by TOPS_1:8;

Lower_Arc C is_an_arc_of W-min C, E-max C by Th50;

then First_Point ((Lower_Arc C),(W-min C),(E-max C),(Vertical_Line (((W-bound C) + (E-bound C)) / 2))) in (Vertical_Line (((W-bound C) + (E-bound C)) / 2)) /\ (Lower_Arc C) by A1, A2, JORDAN5C:def 1;

hence Lower_Middle_Point C in Lower_Arc C by XBOOLE_0:def 4; :: thesis: verum

set L = Vertical_Line (((W-bound C) + (E-bound C)) / 2);

set p = First_Point ((Lower_Arc C),(W-min C),(E-max C),(Vertical_Line (((W-bound C) + (E-bound C)) / 2)));

A1: Lower_Arc C meets Vertical_Line (((W-bound C) + (E-bound C)) / 2) by Th62;

Vertical_Line (((W-bound C) + (E-bound C)) / 2) is closed by Th6;

then A2: (Lower_Arc C) /\ (Vertical_Line (((W-bound C) + (E-bound C)) / 2)) is closed by TOPS_1:8;

Lower_Arc C is_an_arc_of W-min C, E-max C by Th50;

then First_Point ((Lower_Arc C),(W-min C),(E-max C),(Vertical_Line (((W-bound C) + (E-bound C)) / 2))) in (Vertical_Line (((W-bound C) + (E-bound C)) / 2)) /\ (Lower_Arc C) by A1, A2, JORDAN5C:def 1;

hence Lower_Middle_Point C in Lower_Arc C by XBOOLE_0:def 4; :: thesis: verum