let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: for n being Nat st n > 0 holds

(First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 > (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2

let n be Nat; :: thesis: ( n > 0 implies (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 > (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 )

set sr = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2;

set Wmin = W-min (L~ (Cage (C,n)));

set Emax = E-max (L~ (Cage (C,n)));

set Nbo = N-bound (L~ (Cage (C,n)));

set Ebo = E-bound (L~ (Cage (C,n)));

set Sbo = S-bound (L~ (Cage (C,n)));

set Wbo = W-bound (L~ (Cage (C,n)));

set SW = SW-corner (L~ (Cage (C,n)));

set FiP = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)));

set LaP = Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)));

set g = (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ^ <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*>;

set h = (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>;

A1: (Upper_Seq (C,n)) /. 1 = W-min (L~ (Cage (C,n))) by JORDAN1F:5;

A2: W-bound (L~ (Cage (C,n))) <= E-bound (L~ (Cage (C,n))) by SPRECT_1:21;

then W-bound (L~ (Cage (C,n))) <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by JORDAN6:1;

then A3: (W-min (L~ (Cage (C,n)))) `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by EUCLID:52;

((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 <= E-bound (L~ (Cage (C,n))) by A2, JORDAN6:1;

then A4: ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 <= (E-max (L~ (Cage (C,n)))) `1 by EUCLID:52;

set GCw = (Gauge (C,n)) * ((Center (Gauge (C,n))),(width (Gauge (C,n))));

A5: 1 <= Center (Gauge (C,n)) by JORDAN1B:11;

len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def 1;

then A6: ((Gauge (C,n)) * ((Center (Gauge (C,n))),(width (Gauge (C,n))))) `2 = N-bound (L~ (Cage (C,n))) by A5, JORDAN1A:70, JORDAN1B:13;

A7: (SW-corner (L~ (Cage (C,n)))) `2 <= (W-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:30;

A8: |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| `2 = N-bound (L~ (Cage (C,n))) by EUCLID:52;

set RevL = (Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))));

A9: ( <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> is one-to-one & <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> is special ) by FINSEQ_3:93;

A10: rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) c= rng (Rev (Lower_Seq (C,n))) by FINSEQ_5:48;

A11: ( (Lower_Seq (C,n)) /. 1 = E-max (L~ (Cage (C,n))) & (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = W-min (L~ (Cage (C,n))) ) by JORDAN1F:6, JORDAN1F:8;

then A12: L~ (Lower_Seq (C,n)) is_an_arc_of E-max (L~ (Cage (C,n))), W-min (L~ (Cage (C,n))) by TOPREAL1:25;

A13: 4 <= len (Gauge (C,n)) by JORDAN8:10;

then A14: len (Gauge (C,n)) >= 3 by XXREAL_0:2;

A15: W-bound (L~ (Cage (C,n))) < E-bound (L~ (Cage (C,n))) by SPRECT_1:31;

then A16: W-bound (L~ (Cage (C,n))) < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by XREAL_1:226;

L~ (Lower_Seq (C,n)) is_an_arc_of W-min (L~ (Cage (C,n))), E-max (L~ (Cage (C,n))) by A11, JORDAN5B:14, TOPREAL1:25;

then A17: ( L~ (Lower_Seq (C,n)) meets Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) & (L~ (Lower_Seq (C,n))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) is closed ) by A3, A4, JORDAN6:49;

then A18: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in (L~ (Lower_Seq (C,n))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) by A12, JORDAN5C:def 2;

then A19: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in L~ (Lower_Seq (C,n)) by XBOOLE_0:def 4;

then Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by XBOOLE_0:def 3;

then A20: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in L~ (Cage (C,n)) by JORDAN1E:13;

assume A21: n > 0 ; :: thesis: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 > (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2

then A22: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n)) by Th47;

then A23: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_4:20;

then A24: 1 <= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) by FINSEQ_3:25;

1 <= len (Gauge (C,n)) by A13, XXREAL_0:2;

then 1 <= width (Gauge (C,n)) by JORDAN8:def 1;

then ((Gauge (C,n)) * ((Center (Gauge (C,n))),(width (Gauge (C,n))))) `1 = ((W-bound C) + (E-bound C)) / 2 by A21, Th35

.= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by Th33 ;

then (Gauge (C,n)) * ((Center (Gauge (C,n))),(width (Gauge (C,n)))) = |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| by A6, EUCLID:53;

then not |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| in rng (Lower_Seq (C,n)) by A5, A14, Th43, JORDAN1B:15;

then not |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| in rng (Rev (Lower_Seq (C,n))) by FINSEQ_5:57;

then A25: not |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| in rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) by A10;

(SW-corner (L~ (Cage (C,n)))) `2 = S-bound (L~ (Cage (C,n))) by EUCLID:52;

then |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| <> SW-corner (L~ (Cage (C,n))) by A8, SPRECT_1:32;

then not |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| in {(SW-corner (L~ (Cage (C,n))))} by TARSKI:def 1;

then not |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| in rng <*(SW-corner (L~ (Cage (C,n))))*> by FINSEQ_1:38;

then not |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| in (rng <*(SW-corner (L~ (Cage (C,n))))*>) \/ (rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) by A25, XBOOLE_0:def 3;

then not |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| in rng (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) by FINSEQ_1:31;

then rng (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) misses {|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|} by ZFMISC_1:50;

then (rng (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))))) /\ {|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|} = {} ;

then (rng (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))))) /\ (rng <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>) = {} by FINSEQ_1:38;

then A26: rng (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) misses rng <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> ;

Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Lower_Seq (C,n)) by A21, Th48;

then A27: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Rev (Lower_Seq (C,n))) by FINSEQ_5:57;

then A28: not (Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) is empty by FINSEQ_5:47;

A29: len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) = (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) by A27, FINSEQ_5:42;

A30: (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) = E-max (L~ (Cage (C,n))) by JORDAN1F:7;

then A31: L~ (Upper_Seq (C,n)) is_an_arc_of W-min (L~ (Cage (C,n))), E-max (L~ (Cage (C,n))) by A1, TOPREAL1:25;

A32: ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 < E-bound (L~ (Cage (C,n))) by A15, XREAL_1:226;

then A33: ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 <= (E-max (L~ (Cage (C,n)))) `1 by EUCLID:52;

(W-min (L~ (Cage (C,n)))) `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A16, EUCLID:52;

then ( L~ (Upper_Seq (C,n)) meets Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) & (L~ (Upper_Seq (C,n))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) is closed ) by A31, A33, JORDAN6:49;

then A34: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in (L~ (Upper_Seq (C,n))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) by A31, JORDAN5C:def 1;

then A35: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in L~ (Upper_Seq (C,n)) by XBOOLE_0:def 4;

then First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by XBOOLE_0:def 3;

then A36: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in L~ (Cage (C,n)) by JORDAN1E:13;

A39: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by A34, XBOOLE_0:def 4;

then A40: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by JORDAN6:31;

then A42: rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) misses rng <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> by FINSEQ_1:38;

A43: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) <= len (Upper_Seq (C,n)) by A23, FINSEQ_3:25;

Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by A18, XBOOLE_0:def 4;

then A44: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by JORDAN6:31;

then A46: len (Upper_Seq (C,n)) > 2 by XXREAL_0:2;

then A47: 2 in dom (Upper_Seq (C,n)) by FINSEQ_3:25;

then A48: ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))))) `2 = ((Upper_Seq (C,n)) /. ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))) `2 by A23, SPRECT_2:9

.= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 by A22, FINSEQ_5:38

.= |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| `2 by EUCLID:52

.= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. 1) `2 by FINSEQ_4:16 ;

2 <> (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))

then reconsider g = (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ^ <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> as one-to-one special FinSequence of (TOP-REAL 2) by A42, A48, A9, FINSEQ_3:91, GOBOARD2:8;

mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))) is_in_the_area_of Cage (C,n) by A47, A23, JORDAN1E:17, SPRECT_2:22;

then A49: g is_in_the_area_of Cage (C,n) by A38, SPRECT_2:24;

A50: (g /. (len g)) `1 = (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. (len <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*>)) `1 by SPRECT_3:1

.= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. 1) `1 by FINSEQ_1:39

.= |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| `1 by FINSEQ_4:16

.= E-bound (L~ (Cage (C,n))) by EUCLID:52 ;

A51: 1 <= len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by A47, A23, SPRECT_2:5;

then 1 in dom (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by FINSEQ_3:25;

then (g /. 1) `1 = ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. 1) `1 by FINSEQ_4:68

.= ((Upper_Seq (C,n)) /. 2) `1 by A47, A23, SPRECT_2:8

.= W-bound (L~ (Cage (C,n))) by Th31 ;

then A52: g is_a_h.c._for Cage (C,n) by A49, A50, SPRECT_2:def 2;

assume (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 <= (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 ; :: thesis: contradiction

then A53: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 < (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 by A45, XXREAL_0:1;

A54: rng (Lower_Seq (C,n)) c= rng (Cage (C,n)) by Th39;

(First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 > (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2

let n be Nat; :: thesis: ( n > 0 implies (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 > (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 )

set sr = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2;

set Wmin = W-min (L~ (Cage (C,n)));

set Emax = E-max (L~ (Cage (C,n)));

set Nbo = N-bound (L~ (Cage (C,n)));

set Ebo = E-bound (L~ (Cage (C,n)));

set Sbo = S-bound (L~ (Cage (C,n)));

set Wbo = W-bound (L~ (Cage (C,n)));

set SW = SW-corner (L~ (Cage (C,n)));

set FiP = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)));

set LaP = Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)));

set g = (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ^ <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*>;

set h = (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>;

A1: (Upper_Seq (C,n)) /. 1 = W-min (L~ (Cage (C,n))) by JORDAN1F:5;

A2: W-bound (L~ (Cage (C,n))) <= E-bound (L~ (Cage (C,n))) by SPRECT_1:21;

then W-bound (L~ (Cage (C,n))) <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by JORDAN6:1;

then A3: (W-min (L~ (Cage (C,n)))) `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by EUCLID:52;

((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 <= E-bound (L~ (Cage (C,n))) by A2, JORDAN6:1;

then A4: ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 <= (E-max (L~ (Cage (C,n)))) `1 by EUCLID:52;

set GCw = (Gauge (C,n)) * ((Center (Gauge (C,n))),(width (Gauge (C,n))));

A5: 1 <= Center (Gauge (C,n)) by JORDAN1B:11;

len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def 1;

then A6: ((Gauge (C,n)) * ((Center (Gauge (C,n))),(width (Gauge (C,n))))) `2 = N-bound (L~ (Cage (C,n))) by A5, JORDAN1A:70, JORDAN1B:13;

A7: (SW-corner (L~ (Cage (C,n)))) `2 <= (W-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:30;

A8: |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| `2 = N-bound (L~ (Cage (C,n))) by EUCLID:52;

set RevL = (Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))));

A9: ( <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> is one-to-one & <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> is special ) by FINSEQ_3:93;

A10: rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) c= rng (Rev (Lower_Seq (C,n))) by FINSEQ_5:48;

A11: ( (Lower_Seq (C,n)) /. 1 = E-max (L~ (Cage (C,n))) & (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = W-min (L~ (Cage (C,n))) ) by JORDAN1F:6, JORDAN1F:8;

then A12: L~ (Lower_Seq (C,n)) is_an_arc_of E-max (L~ (Cage (C,n))), W-min (L~ (Cage (C,n))) by TOPREAL1:25;

A13: 4 <= len (Gauge (C,n)) by JORDAN8:10;

then A14: len (Gauge (C,n)) >= 3 by XXREAL_0:2;

A15: W-bound (L~ (Cage (C,n))) < E-bound (L~ (Cage (C,n))) by SPRECT_1:31;

then A16: W-bound (L~ (Cage (C,n))) < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by XREAL_1:226;

L~ (Lower_Seq (C,n)) is_an_arc_of W-min (L~ (Cage (C,n))), E-max (L~ (Cage (C,n))) by A11, JORDAN5B:14, TOPREAL1:25;

then A17: ( L~ (Lower_Seq (C,n)) meets Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) & (L~ (Lower_Seq (C,n))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) is closed ) by A3, A4, JORDAN6:49;

then A18: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in (L~ (Lower_Seq (C,n))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) by A12, JORDAN5C:def 2;

then A19: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in L~ (Lower_Seq (C,n)) by XBOOLE_0:def 4;

then Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by XBOOLE_0:def 3;

then A20: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in L~ (Cage (C,n)) by JORDAN1E:13;

assume A21: n > 0 ; :: thesis: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 > (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2

then A22: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n)) by Th47;

then A23: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_4:20;

then A24: 1 <= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) by FINSEQ_3:25;

1 <= len (Gauge (C,n)) by A13, XXREAL_0:2;

then 1 <= width (Gauge (C,n)) by JORDAN8:def 1;

then ((Gauge (C,n)) * ((Center (Gauge (C,n))),(width (Gauge (C,n))))) `1 = ((W-bound C) + (E-bound C)) / 2 by A21, Th35

.= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by Th33 ;

then (Gauge (C,n)) * ((Center (Gauge (C,n))),(width (Gauge (C,n)))) = |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| by A6, EUCLID:53;

then not |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| in rng (Lower_Seq (C,n)) by A5, A14, Th43, JORDAN1B:15;

then not |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| in rng (Rev (Lower_Seq (C,n))) by FINSEQ_5:57;

then A25: not |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| in rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) by A10;

(SW-corner (L~ (Cage (C,n)))) `2 = S-bound (L~ (Cage (C,n))) by EUCLID:52;

then |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| <> SW-corner (L~ (Cage (C,n))) by A8, SPRECT_1:32;

then not |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| in {(SW-corner (L~ (Cage (C,n))))} by TARSKI:def 1;

then not |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| in rng <*(SW-corner (L~ (Cage (C,n))))*> by FINSEQ_1:38;

then not |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| in (rng <*(SW-corner (L~ (Cage (C,n))))*>) \/ (rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) by A25, XBOOLE_0:def 3;

then not |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| in rng (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) by FINSEQ_1:31;

then rng (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) misses {|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|} by ZFMISC_1:50;

then (rng (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))))) /\ {|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|} = {} ;

then (rng (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))))) /\ (rng <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>) = {} by FINSEQ_1:38;

then A26: rng (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) misses rng <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> ;

Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Lower_Seq (C,n)) by A21, Th48;

then A27: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Rev (Lower_Seq (C,n))) by FINSEQ_5:57;

then A28: not (Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) is empty by FINSEQ_5:47;

A29: len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) = (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) by A27, FINSEQ_5:42;

A30: (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) = E-max (L~ (Cage (C,n))) by JORDAN1F:7;

then A31: L~ (Upper_Seq (C,n)) is_an_arc_of W-min (L~ (Cage (C,n))), E-max (L~ (Cage (C,n))) by A1, TOPREAL1:25;

A32: ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 < E-bound (L~ (Cage (C,n))) by A15, XREAL_1:226;

then A33: ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 <= (E-max (L~ (Cage (C,n)))) `1 by EUCLID:52;

(W-min (L~ (Cage (C,n)))) `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A16, EUCLID:52;

then ( L~ (Upper_Seq (C,n)) meets Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) & (L~ (Upper_Seq (C,n))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) is closed ) by A31, A33, JORDAN6:49;

then A34: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in (L~ (Upper_Seq (C,n))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) by A31, JORDAN5C:def 1;

then A35: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in L~ (Upper_Seq (C,n)) by XBOOLE_0:def 4;

then First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by XBOOLE_0:def 3;

then A36: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in L~ (Cage (C,n)) by JORDAN1E:13;

now :: thesis: for m being Nat st m in dom <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> holds

( W-bound (L~ (Cage (C,n))) <= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `1 & (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 & (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )

then A38:
<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> is_in_the_area_of Cage (C,n)
by SPRECT_2:def 1;( W-bound (L~ (Cage (C,n))) <= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `1 & (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 & (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )

let m be Nat; :: thesis: ( m in dom <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> implies ( W-bound (L~ (Cage (C,n))) <= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `1 & (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 & (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) )

assume m in dom <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> ; :: thesis: ( W-bound (L~ (Cage (C,n))) <= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `1 & (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 & (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )

then m in Seg 1 by FINSEQ_1:38;

then m = 1 by FINSEQ_1:2, TARSKI:def 1;

then A37: <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m = |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| by FINSEQ_4:16;

then (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52;

hence ( W-bound (L~ (Cage (C,n))) <= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `1 & (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) ) by SPRECT_1:21; :: thesis: ( S-bound (L~ (Cage (C,n))) <= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 & (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )

(<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 = (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 by A37, EUCLID:52;

hence ( S-bound (L~ (Cage (C,n))) <= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 & (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) by A36, PSCOMP_1:24; :: thesis: verum

end;assume m in dom <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> ; :: thesis: ( W-bound (L~ (Cage (C,n))) <= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `1 & (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 & (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )

then m in Seg 1 by FINSEQ_1:38;

then m = 1 by FINSEQ_1:2, TARSKI:def 1;

then A37: <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m = |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| by FINSEQ_4:16;

then (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52;

hence ( W-bound (L~ (Cage (C,n))) <= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `1 & (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) ) by SPRECT_1:21; :: thesis: ( S-bound (L~ (Cage (C,n))) <= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 & (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )

(<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 = (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 by A37, EUCLID:52;

hence ( S-bound (L~ (Cage (C,n))) <= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 & (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) by A36, PSCOMP_1:24; :: thesis: verum

A39: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by A34, XBOOLE_0:def 4;

then A40: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by JORDAN6:31;

now :: thesis: not (rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))))) /\ {|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|} <> {}

then
rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) misses {|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|}
;assume
(rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))))) /\ {|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|} <> {}
; :: thesis: contradiction

then consider x being object such that

A41: x in (rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))))) /\ {|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|} by XBOOLE_0:def 1;

( x in rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) & x in {|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|} ) by A41, XBOOLE_0:def 4;

then |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| in rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by TARSKI:def 1;

then |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A21, Th53;

hence contradiction by A32, EUCLID:52; :: thesis: verum

end;then consider x being object such that

A41: x in (rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))))) /\ {|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|} by XBOOLE_0:def 1;

( x in rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) & x in {|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|} ) by A41, XBOOLE_0:def 4;

then |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| in rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by TARSKI:def 1;

then |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A21, Th53;

hence contradiction by A32, EUCLID:52; :: thesis: verum

then A42: rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) misses rng <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> by FINSEQ_1:38;

A43: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) <= len (Upper_Seq (C,n)) by A23, FINSEQ_3:25;

Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by A18, XBOOLE_0:def 4;

then A44: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by JORDAN6:31;

A45: now :: thesis: not (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 = (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2

len (Upper_Seq (C,n)) >= 3
by JORDAN1E:15;assume
(First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 = (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2
; :: thesis: contradiction

then First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A40, A44, TOPREAL3:6;

then First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A35, A19, XBOOLE_0:def 4;

then First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;

then ( First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = W-min (L~ (Cage (C,n))) or First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = E-max (L~ (Cage (C,n))) ) by TARSKI:def 2;

hence contradiction by A16, A32, A40, EUCLID:52; :: thesis: verum

end;then First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A40, A44, TOPREAL3:6;

then First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A35, A19, XBOOLE_0:def 4;

then First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;

then ( First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = W-min (L~ (Cage (C,n))) or First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = E-max (L~ (Cage (C,n))) ) by TARSKI:def 2;

hence contradiction by A16, A32, A40, EUCLID:52; :: thesis: verum

then A46: len (Upper_Seq (C,n)) > 2 by XXREAL_0:2;

then A47: 2 in dom (Upper_Seq (C,n)) by FINSEQ_3:25;

then A48: ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))))) `2 = ((Upper_Seq (C,n)) /. ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))) `2 by A23, SPRECT_2:9

.= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 by A22, FINSEQ_5:38

.= |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| `2 by EUCLID:52

.= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. 1) `2 by FINSEQ_4:16 ;

2 <> (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))

proof

then
mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))) is being_S-Seq
by A46, A24, A43, JORDAN3:6;
assume
2 = (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))
; :: thesis: contradiction

then (Upper_Seq (C,n)) /. 2 = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A22, FINSEQ_5:38;

then (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 = W-bound (L~ (Cage (C,n))) by Th31;

then W-bound (L~ (Cage (C,n))) = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A39, JORDAN6:31;

hence contradiction by SPRECT_1:31; :: thesis: verum

end;then (Upper_Seq (C,n)) /. 2 = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A22, FINSEQ_5:38;

then (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 = W-bound (L~ (Cage (C,n))) by Th31;

then W-bound (L~ (Cage (C,n))) = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A39, JORDAN6:31;

hence contradiction by SPRECT_1:31; :: thesis: verum

then reconsider g = (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ^ <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> as one-to-one special FinSequence of (TOP-REAL 2) by A42, A48, A9, FINSEQ_3:91, GOBOARD2:8;

mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))) is_in_the_area_of Cage (C,n) by A47, A23, JORDAN1E:17, SPRECT_2:22;

then A49: g is_in_the_area_of Cage (C,n) by A38, SPRECT_2:24;

A50: (g /. (len g)) `1 = (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. (len <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*>)) `1 by SPRECT_3:1

.= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. 1) `1 by FINSEQ_1:39

.= |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| `1 by FINSEQ_4:16

.= E-bound (L~ (Cage (C,n))) by EUCLID:52 ;

A51: 1 <= len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by A47, A23, SPRECT_2:5;

then 1 in dom (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by FINSEQ_3:25;

then (g /. 1) `1 = ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. 1) `1 by FINSEQ_4:68

.= ((Upper_Seq (C,n)) /. 2) `1 by A47, A23, SPRECT_2:8

.= W-bound (L~ (Cage (C,n))) by Th31 ;

then A52: g is_a_h.c._for Cage (C,n) by A49, A50, SPRECT_2:def 2;

assume (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 <= (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 ; :: thesis: contradiction

then A53: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 < (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 by A45, XXREAL_0:1;

A54: rng (Lower_Seq (C,n)) c= rng (Cage (C,n)) by Th39;

now :: thesis: contradiction

per cases
( SW-corner (L~ (Cage (C,n))) <> W-min (L~ (Cage (C,n))) or SW-corner (L~ (Cage (C,n))) = W-min (L~ (Cage (C,n))) )
;

suppose A55:
SW-corner (L~ (Cage (C,n))) <> W-min (L~ (Cage (C,n)))
; :: thesis: contradiction

not SW-corner (L~ (Cage (C,n))) in rng (Lower_Seq (C,n))

then not SW-corner (L~ (Cage (C,n))) in rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) by A10;

then {(SW-corner (L~ (Cage (C,n))))} misses rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) by ZFMISC_1:50;

then {(SW-corner (L~ (Cage (C,n))))} /\ (rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) = {} ;

then (rng <*(SW-corner (L~ (Cage (C,n))))*>) /\ (rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) = {} by FINSEQ_1:38;

then A59: rng <*(SW-corner (L~ (Cage (C,n))))*> misses rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ;

<*(SW-corner (L~ (Cage (C,n))))*> is one-to-one by FINSEQ_3:93;

then A60: <*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) is one-to-one by A59, FINSEQ_3:91;

set FiP2 = First_Point ((L~ (Lower_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)));

set midU = mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))));

reconsider RevLS = Rev (Lower_Seq (C,n)) as special FinSequence of (TOP-REAL 2) ;

(<*(SW-corner (L~ (Cage (C,n))))*> /. (len <*(SW-corner (L~ (Cage (C,n))))*>)) `1 = (<*(SW-corner (L~ (Cage (C,n))))*> /. 1) `1 by FINSEQ_1:39

.= (SW-corner (L~ (Cage (C,n)))) `1 by FINSEQ_4:16

.= W-bound (L~ (Cage (C,n))) by EUCLID:52

.= (W-min (L~ (Cage (C,n)))) `1 by EUCLID:52

.= ((Lower_Seq (C,n)) /. (len (Lower_Seq (C,n)))) `1 by JORDAN1F:8

.= ((Rev (Lower_Seq (C,n))) /. 1) `1 by FINSEQ_5:65

.= (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. 1) `1 by A27, FINSEQ_5:44 ;

then A61: <*(SW-corner (L~ (Cage (C,n))))*> ^ (RevLS -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) is special by GOBOARD2:8;

not (Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) is empty by A27, FINSEQ_5:47;

then A62: ((<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) /. (len (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))))) `1 = (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))))) `1 by SPRECT_3:1

.= (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))))) `1 by A27, FINSEQ_5:42

.= (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 by A27, FINSEQ_5:45

.= |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| `1 by A44, EUCLID:52

.= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. 1) `1 by FINSEQ_4:16 ;

( <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> is one-to-one & <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> is special ) by FINSEQ_3:93;

then reconsider h = (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> as one-to-one special FinSequence of (TOP-REAL 2) by A26, A60, A62, A61, FINSEQ_3:91, GOBOARD2:8;

A63: |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52;

A66: ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) = ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))) by A27, FINSEQ_5:42

.= Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A27, FINSEQ_5:45 ;

A70: ( L~ (Rev (Lower_Seq (C,n))) = L~ (Lower_Seq (C,n)) & First_Point ((L~ (Lower_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) ) by A12, A17, JORDAN5C:18, SPPOL_2:22;

Rev (Lower_Seq (C,n)) is_in_the_area_of Cage (C,n) by JORDAN1E:18, SPRECT_3:51;

then (Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) is_in_the_area_of Cage (C,n) by A27, JORDAN1E:1;

then <*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) is_in_the_area_of Cage (C,n) by A65, SPRECT_2:24;

then A71: h is_in_the_area_of Cage (C,n) by A69, SPRECT_2:24;

len (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) = 1 + (len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) by FINSEQ_5:8;

then A72: len (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) >= 1 by NAT_1:11;

1 in dom h by FINSEQ_5:6;

then h /. 1 = h . 1 by PARTFUN1:def 6;

then A73: (h /. 1) `2 = ((<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) /. 1) `2 by A72, FINSEQ_6:109

.= (SW-corner (L~ (Cage (C,n)))) `2 by FINSEQ_5:15

.= S-bound (L~ (Cage (C,n))) by EUCLID:52 ;

A74: len h = (len (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))))) + 1 by FINSEQ_2:16;

then A75: 1 + 1 <= len h by A72, XREAL_1:7;

L~ (Cage (C,n)) = (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by JORDAN1E:13;

then A76: L~ (Upper_Seq (C,n)) c= L~ (Cage (C,n)) by XBOOLE_1:7;

A77: (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))))) = (Upper_Seq (C,n)) /. ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) by A47, A23, SPRECT_2:9

.= First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A22, FINSEQ_5:38 ;

A78: W-min (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) by A1, FINSEQ_6:42;

then A79: (1 + 1) + 0 <= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) by NAT_1:13;

then ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 2 >= 0 by XREAL_1:19;

then ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) -' 2 = ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 2 by XREAL_0:def 2;

then A80: len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) = (((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 2) + 1 by A43, A79, JORDAN4:8

.= ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - (2 - 1) ;

1 in dom ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) by A28, FINSEQ_5:6;

then A81: (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>) /. 1 = ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. 1 by FINSEQ_4:68

.= (Rev (Lower_Seq (C,n))) /. 1 by A27, FINSEQ_5:44

.= (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by FINSEQ_5:65

.= W-min (L~ (Cage (C,n))) by JORDAN1F:8 ;

A82: (SW-corner (L~ (Cage (C,n)))) `2 <= (W-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:30;

len g = (len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))))) + 1 by FINSEQ_2:16;

then A83: 1 + 1 <= len g by A51, XREAL_1:7;

A84: L~ g = (L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))))) \/ (LSeg (((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|)) by A47, A23, SPPOL_2:19, SPRECT_2:7;

L~ (Rev (Lower_Seq (C,n))) = (L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) \/ (L~ ((Rev (Lower_Seq (C,n))) :- (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) by A27, SPPOL_2:24;

then L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) c= L~ (Rev (Lower_Seq (C,n))) by XBOOLE_1:7;

then A85: L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) c= L~ (Lower_Seq (C,n)) by SPPOL_2:22;

A86: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 <= N-bound (L~ (Cage (C,n))) by A20, PSCOMP_1:24;

A87: |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| `2 = (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 by EUCLID:52;

then A88: LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) is horizontal by SPPOL_1:15;

(Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 = |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| `1 by A44, EUCLID:52;

then A89: LSeg ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|) is vertical by SPPOL_1:16;

A90: L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) c= L~ (Upper_Seq (C,n)) by A47, A23, SPRECT_3:18;

(h /. (len h)) `2 = |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| `2 by A74, FINSEQ_4:67

.= N-bound (L~ (Cage (C,n))) by EUCLID:52 ;

then h is_a_v.c._for Cage (C,n) by A71, A73, SPRECT_2:def 3;

then L~ g meets L~ h by A52, A75, A83, SPRECT_2:29;

then consider x being object such that

A91: x in L~ g and

A92: x in L~ h by XBOOLE_0:3;

reconsider x = x as Point of (TOP-REAL 2) by A91;

L~ h = L~ (<*(SW-corner (L~ (Cage (C,n))))*> ^ (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>)) by FINSEQ_1:32

.= (LSeg ((SW-corner (L~ (Cage (C,n)))),((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>) /. 1))) \/ (L~ (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>)) by SPPOL_2:20

.= (LSeg ((SW-corner (L~ (Cage (C,n)))),((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>) /. 1))) \/ ((L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) \/ (LSeg ((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))))),|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|))) by A27, FINSEQ_5:47, SPPOL_2:19 ;

then A93: ( x in LSeg ((SW-corner (L~ (Cage (C,n)))),((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>) /. 1)) or x in (L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) \/ (LSeg ((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))))),|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|)) ) by A92, XBOOLE_0:def 3;

A94: (SW-corner (L~ (Cage (C,n)))) `1 = (W-min (L~ (Cage (C,n)))) `1 by PSCOMP_1:29;

then A95: LSeg ((SW-corner (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))) is vertical by SPPOL_1:16;

proof

then
not SW-corner (L~ (Cage (C,n))) in rng (Rev (Lower_Seq (C,n)))
by FINSEQ_5:57;
(SW-corner (L~ (Cage (C,n)))) `1 = (W-min (L~ (Cage (C,n)))) `1
by PSCOMP_1:29;

then A56: (SW-corner (L~ (Cage (C,n)))) `2 <> (W-min (L~ (Cage (C,n)))) `2 by A55, TOPREAL3:6;

assume SW-corner (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) ; :: thesis: contradiction

then A57: SW-corner (L~ (Cage (C,n))) in rng (Cage (C,n)) by A54;

len (Cage (C,n)) > 4 by GOBOARD7:34;

then A58: rng (Cage (C,n)) c= L~ (Cage (C,n)) by SPPOL_2:18, XXREAL_0:2;

(SW-corner (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52;

then SW-corner (L~ (Cage (C,n))) in W-most (L~ (Cage (C,n))) by A57, A58, SPRECT_2:12;

then (W-min (L~ (Cage (C,n)))) `2 <= (SW-corner (L~ (Cage (C,n)))) `2 by PSCOMP_1:31;

hence contradiction by A7, A56, XXREAL_0:1; :: thesis: verum

end;then A56: (SW-corner (L~ (Cage (C,n)))) `2 <> (W-min (L~ (Cage (C,n)))) `2 by A55, TOPREAL3:6;

assume SW-corner (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) ; :: thesis: contradiction

then A57: SW-corner (L~ (Cage (C,n))) in rng (Cage (C,n)) by A54;

len (Cage (C,n)) > 4 by GOBOARD7:34;

then A58: rng (Cage (C,n)) c= L~ (Cage (C,n)) by SPPOL_2:18, XXREAL_0:2;

(SW-corner (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52;

then SW-corner (L~ (Cage (C,n))) in W-most (L~ (Cage (C,n))) by A57, A58, SPRECT_2:12;

then (W-min (L~ (Cage (C,n)))) `2 <= (SW-corner (L~ (Cage (C,n)))) `2 by PSCOMP_1:31;

hence contradiction by A7, A56, XXREAL_0:1; :: thesis: verum

then not SW-corner (L~ (Cage (C,n))) in rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) by A10;

then {(SW-corner (L~ (Cage (C,n))))} misses rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) by ZFMISC_1:50;

then {(SW-corner (L~ (Cage (C,n))))} /\ (rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) = {} ;

then (rng <*(SW-corner (L~ (Cage (C,n))))*>) /\ (rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) = {} by FINSEQ_1:38;

then A59: rng <*(SW-corner (L~ (Cage (C,n))))*> misses rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ;

<*(SW-corner (L~ (Cage (C,n))))*> is one-to-one by FINSEQ_3:93;

then A60: <*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) is one-to-one by A59, FINSEQ_3:91;

set FiP2 = First_Point ((L~ (Lower_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)));

set midU = mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))));

reconsider RevLS = Rev (Lower_Seq (C,n)) as special FinSequence of (TOP-REAL 2) ;

(<*(SW-corner (L~ (Cage (C,n))))*> /. (len <*(SW-corner (L~ (Cage (C,n))))*>)) `1 = (<*(SW-corner (L~ (Cage (C,n))))*> /. 1) `1 by FINSEQ_1:39

.= (SW-corner (L~ (Cage (C,n)))) `1 by FINSEQ_4:16

.= W-bound (L~ (Cage (C,n))) by EUCLID:52

.= (W-min (L~ (Cage (C,n)))) `1 by EUCLID:52

.= ((Lower_Seq (C,n)) /. (len (Lower_Seq (C,n)))) `1 by JORDAN1F:8

.= ((Rev (Lower_Seq (C,n))) /. 1) `1 by FINSEQ_5:65

.= (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. 1) `1 by A27, FINSEQ_5:44 ;

then A61: <*(SW-corner (L~ (Cage (C,n))))*> ^ (RevLS -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) is special by GOBOARD2:8;

not (Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) is empty by A27, FINSEQ_5:47;

then A62: ((<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) /. (len (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))))) `1 = (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))))) `1 by SPRECT_3:1

.= (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))))) `1 by A27, FINSEQ_5:42

.= (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 by A27, FINSEQ_5:45

.= |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| `1 by A44, EUCLID:52

.= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. 1) `1 by FINSEQ_4:16 ;

( <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> is one-to-one & <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> is special ) by FINSEQ_3:93;

then reconsider h = (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> as one-to-one special FinSequence of (TOP-REAL 2) by A26, A60, A62, A61, FINSEQ_3:91, GOBOARD2:8;

A63: |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52;

now :: thesis: for m being Nat st m in dom <*(SW-corner (L~ (Cage (C,n))))*> holds

( W-bound (L~ (Cage (C,n))) <= (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `1 & (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 & (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )

then A65:
<*(SW-corner (L~ (Cage (C,n))))*> is_in_the_area_of Cage (C,n)
by SPRECT_2:def 1;( W-bound (L~ (Cage (C,n))) <= (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `1 & (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 & (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )

let m be Nat; :: thesis: ( m in dom <*(SW-corner (L~ (Cage (C,n))))*> implies ( W-bound (L~ (Cage (C,n))) <= (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `1 & (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 & (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) )

assume m in dom <*(SW-corner (L~ (Cage (C,n))))*> ; :: thesis: ( W-bound (L~ (Cage (C,n))) <= (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `1 & (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 & (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )

then m in Seg 1 by FINSEQ_1:38;

then m = 1 by FINSEQ_1:2, TARSKI:def 1;

then A64: <*(SW-corner (L~ (Cage (C,n))))*> /. m = SW-corner (L~ (Cage (C,n))) by FINSEQ_4:16;

then (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52;

hence ( W-bound (L~ (Cage (C,n))) <= (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `1 & (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) ) by SPRECT_1:21; :: thesis: ( S-bound (L~ (Cage (C,n))) <= (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 & (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )

(<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 = S-bound (L~ (Cage (C,n))) by A64, EUCLID:52;

hence ( S-bound (L~ (Cage (C,n))) <= (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 & (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) by SPRECT_1:22; :: thesis: verum

end;assume m in dom <*(SW-corner (L~ (Cage (C,n))))*> ; :: thesis: ( W-bound (L~ (Cage (C,n))) <= (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `1 & (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 & (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )

then m in Seg 1 by FINSEQ_1:38;

then m = 1 by FINSEQ_1:2, TARSKI:def 1;

then A64: <*(SW-corner (L~ (Cage (C,n))))*> /. m = SW-corner (L~ (Cage (C,n))) by FINSEQ_4:16;

then (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52;

hence ( W-bound (L~ (Cage (C,n))) <= (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `1 & (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) ) by SPRECT_1:21; :: thesis: ( S-bound (L~ (Cage (C,n))) <= (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 & (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )

(<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 = S-bound (L~ (Cage (C,n))) by A64, EUCLID:52;

hence ( S-bound (L~ (Cage (C,n))) <= (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 & (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) by SPRECT_1:22; :: thesis: verum

A66: ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) = ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))) by A27, FINSEQ_5:42

.= Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A27, FINSEQ_5:45 ;

now :: thesis: for m being Nat st m in dom <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> holds

( W-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )

then A69:
<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> is_in_the_area_of Cage (C,n)
by SPRECT_2:def 1;( W-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )

let m be Nat; :: thesis: ( m in dom <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> implies ( W-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) )

A67: W-bound (L~ (Cage (C,n))) <= E-bound (L~ (Cage (C,n))) by SPRECT_1:21;

assume m in dom <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> ; :: thesis: ( W-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )

then m in Seg 1 by FINSEQ_1:38;

then m = 1 by FINSEQ_1:2, TARSKI:def 1;

then A68: <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m = |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| by FINSEQ_4:16;

then (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by EUCLID:52;

hence ( W-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) ) by A67, JORDAN6:1; :: thesis: ( S-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )

(<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 = N-bound (L~ (Cage (C,n))) by A68, EUCLID:52;

hence ( S-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) by SPRECT_1:22; :: thesis: verum

end;A67: W-bound (L~ (Cage (C,n))) <= E-bound (L~ (Cage (C,n))) by SPRECT_1:21;

assume m in dom <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> ; :: thesis: ( W-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )

then m in Seg 1 by FINSEQ_1:38;

then m = 1 by FINSEQ_1:2, TARSKI:def 1;

then A68: <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m = |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| by FINSEQ_4:16;

then (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by EUCLID:52;

hence ( W-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) ) by A67, JORDAN6:1; :: thesis: ( S-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )

(<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 = N-bound (L~ (Cage (C,n))) by A68, EUCLID:52;

hence ( S-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) by SPRECT_1:22; :: thesis: verum

A70: ( L~ (Rev (Lower_Seq (C,n))) = L~ (Lower_Seq (C,n)) & First_Point ((L~ (Lower_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) ) by A12, A17, JORDAN5C:18, SPPOL_2:22;

Rev (Lower_Seq (C,n)) is_in_the_area_of Cage (C,n) by JORDAN1E:18, SPRECT_3:51;

then (Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) is_in_the_area_of Cage (C,n) by A27, JORDAN1E:1;

then <*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) is_in_the_area_of Cage (C,n) by A65, SPRECT_2:24;

then A71: h is_in_the_area_of Cage (C,n) by A69, SPRECT_2:24;

len (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) = 1 + (len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) by FINSEQ_5:8;

then A72: len (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) >= 1 by NAT_1:11;

1 in dom h by FINSEQ_5:6;

then h /. 1 = h . 1 by PARTFUN1:def 6;

then A73: (h /. 1) `2 = ((<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) /. 1) `2 by A72, FINSEQ_6:109

.= (SW-corner (L~ (Cage (C,n)))) `2 by FINSEQ_5:15

.= S-bound (L~ (Cage (C,n))) by EUCLID:52 ;

A74: len h = (len (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))))) + 1 by FINSEQ_2:16;

then A75: 1 + 1 <= len h by A72, XREAL_1:7;

L~ (Cage (C,n)) = (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by JORDAN1E:13;

then A76: L~ (Upper_Seq (C,n)) c= L~ (Cage (C,n)) by XBOOLE_1:7;

A77: (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))))) = (Upper_Seq (C,n)) /. ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) by A47, A23, SPRECT_2:9

.= First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A22, FINSEQ_5:38 ;

A78: W-min (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) by A1, FINSEQ_6:42;

now :: thesis: not (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) = 1

then
(First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) > 1
by A24, XXREAL_0:1;assume
(First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) = 1
; :: thesis: contradiction

then (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) = ((Upper_Seq (C,n)) /. 1) .. (Upper_Seq (C,n)) by FINSEQ_6:43

.= (W-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) by JORDAN1F:5 ;

then First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = W-min (L~ (Cage (C,n))) by A22, A78, FINSEQ_5:9;

hence contradiction by A16, A40, EUCLID:52; :: thesis: verum

end;then (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) = ((Upper_Seq (C,n)) /. 1) .. (Upper_Seq (C,n)) by FINSEQ_6:43

.= (W-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) by JORDAN1F:5 ;

then First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = W-min (L~ (Cage (C,n))) by A22, A78, FINSEQ_5:9;

hence contradiction by A16, A40, EUCLID:52; :: thesis: verum

then A79: (1 + 1) + 0 <= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) by NAT_1:13;

then ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 2 >= 0 by XREAL_1:19;

then ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) -' 2 = ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 2 by XREAL_0:def 2;

then A80: len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) = (((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 2) + 1 by A43, A79, JORDAN4:8

.= ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - (2 - 1) ;

1 in dom ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) by A28, FINSEQ_5:6;

then A81: (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>) /. 1 = ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. 1 by FINSEQ_4:68

.= (Rev (Lower_Seq (C,n))) /. 1 by A27, FINSEQ_5:44

.= (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by FINSEQ_5:65

.= W-min (L~ (Cage (C,n))) by JORDAN1F:8 ;

A82: (SW-corner (L~ (Cage (C,n)))) `2 <= (W-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:30;

len g = (len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))))) + 1 by FINSEQ_2:16;

then A83: 1 + 1 <= len g by A51, XREAL_1:7;

A84: L~ g = (L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))))) \/ (LSeg (((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|)) by A47, A23, SPPOL_2:19, SPRECT_2:7;

L~ (Rev (Lower_Seq (C,n))) = (L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) \/ (L~ ((Rev (Lower_Seq (C,n))) :- (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) by A27, SPPOL_2:24;

then L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) c= L~ (Rev (Lower_Seq (C,n))) by XBOOLE_1:7;

then A85: L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) c= L~ (Lower_Seq (C,n)) by SPPOL_2:22;

A86: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 <= N-bound (L~ (Cage (C,n))) by A20, PSCOMP_1:24;

A87: |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| `2 = (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 by EUCLID:52;

then A88: LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) is horizontal by SPPOL_1:15;

(Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 = |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| `1 by A44, EUCLID:52;

then A89: LSeg ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|) is vertical by SPPOL_1:16;

A90: L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) c= L~ (Upper_Seq (C,n)) by A47, A23, SPRECT_3:18;

(h /. (len h)) `2 = |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| `2 by A74, FINSEQ_4:67

.= N-bound (L~ (Cage (C,n))) by EUCLID:52 ;

then h is_a_v.c._for Cage (C,n) by A71, A73, SPRECT_2:def 3;

then L~ g meets L~ h by A52, A75, A83, SPRECT_2:29;

then consider x being object such that

A91: x in L~ g and

A92: x in L~ h by XBOOLE_0:3;

reconsider x = x as Point of (TOP-REAL 2) by A91;

L~ h = L~ (<*(SW-corner (L~ (Cage (C,n))))*> ^ (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>)) by FINSEQ_1:32

.= (LSeg ((SW-corner (L~ (Cage (C,n)))),((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>) /. 1))) \/ (L~ (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>)) by SPPOL_2:20

.= (LSeg ((SW-corner (L~ (Cage (C,n)))),((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>) /. 1))) \/ ((L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) \/ (LSeg ((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))))),|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|))) by A27, FINSEQ_5:47, SPPOL_2:19 ;

then A93: ( x in LSeg ((SW-corner (L~ (Cage (C,n)))),((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>) /. 1)) or x in (L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) \/ (LSeg ((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))))),|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|)) ) by A92, XBOOLE_0:def 3;

A94: (SW-corner (L~ (Cage (C,n)))) `1 = (W-min (L~ (Cage (C,n)))) `1 by PSCOMP_1:29;

then A95: LSeg ((SW-corner (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))) is vertical by SPPOL_1:16;

now :: thesis: contradiction

per cases
( x in LSeg ((SW-corner (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))) or x in L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) or x in LSeg ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|) )
by A93, A81, A66, XBOOLE_0:def 3;

suppose A96:
x in LSeg ((SW-corner (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))))
; :: thesis: contradiction

then A97:
x `2 <= (W-min (L~ (Cage (C,n)))) `2
by A82, TOPREAL1:4;

A98: x `1 = (SW-corner (L~ (Cage (C,n)))) `1 by A95, A96, SPPOL_1:41;

then A99: x `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52;

end;A98: x `1 = (SW-corner (L~ (Cage (C,n)))) `1 by A95, A96, SPPOL_1:41;

then A99: x `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52;

now :: thesis: contradictionend;

hence
contradiction
; :: thesis: verumper cases
( x in L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) or x in LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) )
by A91, A84, A77, XBOOLE_0:def 3;

end;

suppose A100:
x in L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))))
; :: thesis: contradiction

then
x in L~ (Upper_Seq (C,n))
by A90;

then x in W-most (L~ (Cage (C,n))) by A76, A98, EUCLID:52, SPRECT_2:12;

then x `2 >= (W-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:31;

then x `2 = (W-min (L~ (Cage (C,n)))) `2 by A97, XXREAL_0:1;

then x = W-min (L~ (Cage (C,n))) by A94, A98, TOPREAL3:6;

then (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) = 1 by A46, A1, A24, A43, A100, Th37;

then W-min (L~ (Cage (C,n))) = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A1, A22, FINSEQ_5:38;

hence contradiction by A16, A40, EUCLID:52; :: thesis: verum

end;then x in W-most (L~ (Cage (C,n))) by A76, A98, EUCLID:52, SPRECT_2:12;

then x `2 >= (W-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:31;

then x `2 = (W-min (L~ (Cage (C,n)))) `2 by A97, XXREAL_0:1;

then x = W-min (L~ (Cage (C,n))) by A94, A98, TOPREAL3:6;

then (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) = 1 by A46, A1, A24, A43, A100, Th37;

then W-min (L~ (Cage (C,n))) = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A1, A22, FINSEQ_5:38;

hence contradiction by A16, A40, EUCLID:52; :: thesis: verum

suppose
x in LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|)
; :: thesis: contradiction

end;

end;

suppose A101:
x in L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))
; :: thesis: contradiction

now :: thesis: contradiction

per cases
( x in L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) or x in LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) )
by A91, A84, A77, XBOOLE_0:def 3;

suppose A102:
x in L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))))
; :: thesis: contradiction

then
x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n)))
by A90, A85, A101, XBOOLE_0:def 4;

then A103: x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;

then A103: x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;

now :: thesis: contradiction

per cases
( x = W-min (L~ (Cage (C,n))) or x = E-max (L~ (Cage (C,n))) )
by A103, TARSKI:def 2;

suppose
x = W-min (L~ (Cage (C,n)))
; :: thesis: contradiction

then
(First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) = 1
by A46, A1, A24, A43, A102, Th37;

then W-min (L~ (Cage (C,n))) = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A1, A22, FINSEQ_5:38;

hence contradiction by A16, A40, EUCLID:52; :: thesis: verum

end;then W-min (L~ (Cage (C,n))) = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A1, A22, FINSEQ_5:38;

hence contradiction by A16, A40, EUCLID:52; :: thesis: verum

suppose
x = E-max (L~ (Cage (C,n)))
; :: thesis: contradiction

then
(First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) = len (Upper_Seq (C,n))
by A46, A30, A24, A43, A102, Th38;

then E-max (L~ (Cage (C,n))) = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n))))

then E-max (L~ (Cage (C,n))) = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n))))