let C be Simple_closed_curve; :: thesis: (Upper_Middle_Point C) `1 = ((W-bound C) + (E-bound C)) / 2

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

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

A1: Upper_Arc C meets Vertical_Line (((W-bound C) + (E-bound C)) / 2) by Th63;

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

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

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

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

then First_Point ((Upper_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) by XBOOLE_0:def 4;

hence (Upper_Middle_Point C) `1 = ((W-bound C) + (E-bound C)) / 2 by Th31; :: thesis: verum

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

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

A1: Upper_Arc C meets Vertical_Line (((W-bound C) + (E-bound C)) / 2) by Th63;

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

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

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

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

then First_Point ((Upper_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) by XBOOLE_0:def 4;

hence (Upper_Middle_Point C) `1 = ((W-bound C) + (E-bound C)) / 2 by Th31; :: thesis: verum