let P1, P19 be non empty Subset of (TOP-REAL 2); :: thesis: ( P1 is_an_arc_of E-max P, W-min P & (Upper_Arc P) /\ P1 = {(W-min P),(E-max P)} & (Upper_Arc P) \/ P1 = P & (First_Point ((Upper_Arc P),(W-min P),(E-max P),(Vertical_Line (((W-bound P) + (E-bound P)) / 2)))) `2 > (Last_Point (P1,(E-max P),(W-min P),(Vertical_Line (((W-bound P) + (E-bound P)) / 2)))) `2 & P19 is_an_arc_of E-max P, W-min P & (Upper_Arc P) /\ P19 = {(W-min P),(E-max P)} & (Upper_Arc P) \/ P19 = P & (First_Point ((Upper_Arc P),(W-min P),(E-max P),(Vertical_Line (((W-bound P) + (E-bound P)) / 2)))) `2 > (Last_Point (P19,(E-max P),(W-min P),(Vertical_Line (((W-bound P) + (E-bound P)) / 2)))) `2 implies P1 = P19 )

assume that

A3: P1 is_an_arc_of E-max P, W-min P and

(Upper_Arc P) /\ P1 = {(W-min P),(E-max P)} and

A4: (Upper_Arc P) \/ P1 = P and

(First_Point ((Upper_Arc P),(W-min P),(E-max P),(Vertical_Line (((W-bound P) + (E-bound P)) / 2)))) `2 > (Last_Point (P1,(E-max P),(W-min P),(Vertical_Line (((W-bound P) + (E-bound P)) / 2)))) `2 and

A5: P19 is_an_arc_of E-max P, W-min P and

(Upper_Arc P) /\ P19 = {(W-min P),(E-max P)} and

A6: (Upper_Arc P) \/ P19 = P and

(First_Point ((Upper_Arc P),(W-min P),(E-max P),(Vertical_Line (((W-bound P) + (E-bound P)) / 2)))) `2 > (Last_Point (P19,(E-max P),(W-min P),(Vertical_Line (((W-bound P) + (E-bound P)) / 2)))) `2 ; :: thesis: P1 = P19

A7: P1 is_an_arc_of W-min P, E-max P by A3, JORDAN5B:14;

P19 is_an_arc_of W-min P, E-max P by A5, JORDAN5B:14;

then ( P1 = P19 or ( Upper_Arc P = P19 & P1 = Upper_Arc P ) ) by A1, A2, A4, A6, A7, Th48;

hence P1 = P19 ; :: thesis: verum

assume that

A3: P1 is_an_arc_of E-max P, W-min P and

(Upper_Arc P) /\ P1 = {(W-min P),(E-max P)} and

A4: (Upper_Arc P) \/ P1 = P and

(First_Point ((Upper_Arc P),(W-min P),(E-max P),(Vertical_Line (((W-bound P) + (E-bound P)) / 2)))) `2 > (Last_Point (P1,(E-max P),(W-min P),(Vertical_Line (((W-bound P) + (E-bound P)) / 2)))) `2 and

A5: P19 is_an_arc_of E-max P, W-min P and

(Upper_Arc P) /\ P19 = {(W-min P),(E-max P)} and

A6: (Upper_Arc P) \/ P19 = P and

(First_Point ((Upper_Arc P),(W-min P),(E-max P),(Vertical_Line (((W-bound P) + (E-bound P)) / 2)))) `2 > (Last_Point (P19,(E-max P),(W-min P),(Vertical_Line (((W-bound P) + (E-bound P)) / 2)))) `2 ; :: thesis: P1 = P19

A7: P1 is_an_arc_of W-min P, E-max P by A3, JORDAN5B:14;

P19 is_an_arc_of W-min P, E-max P by A5, JORDAN5B:14;

then ( P1 = P19 or ( Upper_Arc P = P19 & P1 = Upper_Arc P ) ) by A1, A2, A4, A6, A7, Th48;

hence P1 = P19 ; :: thesis: verum