let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: for n being Element of NAT st n > 0 holds
for k being Element of NAT st 1 <= k & k < (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) holds
((Upper_Seq C,n) /. k) `1 < ((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2
let n be Element of NAT ; :: thesis: ( n > 0 implies for k being Element of NAT st 1 <= k & k < (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) holds
((Upper_Seq C,n) /. k) `1 < ((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2 )
assume A1: n > 0 ; :: thesis: for k being Element of NAT st 1 <= k & k < (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) holds
((Upper_Seq C,n) /. k) `1 < ((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2
set US = Upper_Seq C,n;
set sr = ((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2;
set Ebo = E-bound (L~ (Cage C,n));
set Wbo = W-bound (L~ (Cage C,n));
set Emax = E-max (L~ (Cage C,n));
set Wmin = W-min (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));
defpred S<sub>1</sub>[ Nat] means ( 1 <= $1 & $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) implies ((Upper_Seq C,n) /. $1) `1 < ((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2 );
A2: W-bound (L~ (Cage C,n)) < E-bound (L~ (Cage C,n)) by SPRECT_1:33;
then A3: W-bound (L~ (Cage C,n)) < ((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2 by XREAL_1:228;
A4: ((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2 < E-bound (L~ (Cage C,n)) by A2, XREAL_1:228;
A5: for k being non empty Nat st S₁[k] holds
S₁[k + 1]

proof

set GC1 = (Gauge C,n) * (Center (Gauge C,n)),1;
let k be non empty Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A6: ( 1 <= k & k < (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) implies ((Upper_Seq C,n) /. k) `1 < ((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2 ) ; :: thesis: S<sub>1</sub>[k + 1]
4 <= len (Gauge C,n) by JORDAN8:13;
then 1 <= len (Gauge C,n) by XXREAL_0:2;
then A7: 1 <= width (Gauge C,n) by JORDAN8:def 1;
then A8: ((Gauge C,n) * (Center (Gauge C,n)),1) `1 = ((W-bound C) + (E-bound C)) / 2 by A1, Th43
.= ((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2 by Th41 ;
A9: k >= 1 by NAT_1:14;
A10: (Upper_Seq C,n) /. (len (Upper_Seq C,n)) = E-max (L~ (Cage C,n)) by JORDAN1F:7;
A11: 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 A1, Th55;
then A12: (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_4:30;
then A13: 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_3:27;
A14: 1 <= Center (Gauge C,n) by JORDAN1B:12;
A15: (Upper_Seq C,n) /. 1 = W-min (L~ (Cage C,n)) by JORDAN1F:5;
reconsider kk = k as Element of NAT by ORDINAL1:def 13;
assume that
A16: 1 <= k + 1 and
A17: k + 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) ; :: thesis: ((Upper_Seq C,n) /. (k + 1)) `1 < ((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2
A18: (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 A12, FINSEQ_3:27;
then A19: k + 1 <= len (Upper_Seq C,n) by A17, XXREAL_0:2;
Upper_Seq C,n is_sequence_on Gauge C,n by Th4;
then consider i1, j1, i2, j2 being Element of NAT such that
A20: [i1,j1] in Indices (Gauge C,n) and
A21: (Upper_Seq C,n) /. kk = (Gauge C,n) * i1,j1 and
A22: [i2,j2] in Indices (Gauge C,n) and
A23: (Upper_Seq C,n) /. (kk + 1) = (Gauge C,n) * i2,j2 and
A24: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A9, A19, JORDAN8:6;
A25: 1 <= i1 by A20, MATRIX_1:39;
A26: ( 1 <= j1 & j1 <= width (Gauge C,n) ) by A20, MATRIX_1:39;
A27: j2 <= width (Gauge C,n) by A22, MATRIX_1:39;
A28: ( 1 <= i2 & 1 <= j2 ) by A22, MATRIX_1:39;
A29: i2 <= len (Gauge C,n) by A22, MATRIX_1:39;
A30: i1 <= len (Gauge C,n) by A20, MATRIX_1:39;
A31: ( Center (Gauge C,n) <= len (Gauge C,n) & i1 + 1 >= 1 ) by JORDAN1B:14, NAT_1:11;

now

per cases ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A24;

suppose ( i1 = i2 & j1 + 1 = j2 ) ; :: thesis: ((Upper_Seq C,n) /. (k + 1)) `1 < ((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2

then ((Upper_Seq C,n) /. k) `1 = ((Gauge C,n) * i2,1) `1 by A21, A25, A30, A26, GOBOARD5:3
.= ((Upper_Seq C,n) /. (k + 1)) `1 by A23, A29, A28, A27, GOBOARD5:3 ;
hence ((Upper_Seq C,n) /. (k + 1)) `1 < ((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2 by A6, A17, NAT_1:13, NAT_1:14; :: thesis: verum

end;

suppose A32: ( i1 + 1 = i2 & j1 = j2 ) ; :: thesis: ((Upper_Seq C,n) /. (k + 1)) `1 < ((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2

A33: now

A34: k + 1 >= 1 + 1 by A9, XREAL_1:9;
len (mid (Upper_Seq C,n),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))) = (((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 A13, A18, JORDAN4:20
.= (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 A13, XREAL_1:237 ;
then A35: rng (mid (Upper_Seq C,n),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))) c= L~ (mid (Upper_Seq C,n),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 A17, A34, SPPOL_2:18, XXREAL_0:2;
A36: (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)) = 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 A11, FINSEQ_5:41;

A37: now

assume (Upper_Seq C,n) /. 1 in Vertical_Line (((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2) ; :: thesis: contradiction
then (W-min (L~ (Cage C,n))) `1 = ((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2 by A15, JORDAN6:34;
hence contradiction by A3, EUCLID:56; :: thesis: verum

end;

A38: ( (W-min (L~ (Cage C,n))) `1 <= ((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 <= (E-max (L~ (Cage C,n))) `1 ) by A3, A4, EUCLID:56;
A39: ( Vertical_Line (((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2) is closed & L~ (Upper_Seq C,n) is_an_arc_of W-min (L~ (Cage C,n)), E-max (L~ (Cage C,n)) ) by A15, A10, JORDAN6:33, TOPREAL1:31;
First_Point (L~ (Upper_Seq C,n)),((Upper_Seq C,n) /. 1),((Upper_Seq C,n) /. (len (Upper_Seq C,n))),(Vertical_Line (((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2)) in rng (Upper_Seq C,n) by A1, A15, A10, Th55;
then A40: (L~ (mid (Upper_Seq C,n),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)))) /\ (Vertical_Line (((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage 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 A15, A10, A39, A38, A37, Th58, JORDAN6:64;
A41: ( mid (Upper_Seq C,n),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)) = (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)) & (Upper_Seq C,n) | (Seg ((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 A13, FINSEQ_1:def 15, FINSEQ_6:122;
assume ((Upper_Seq C,n) /. (k + 1)) `1 = ((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2 ; :: thesis: contradiction
then (Upper_Seq C,n) /. (k + 1) in { p where p is Point of (TOP-REAL 2) : p `1 = ((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2 } ;
then A42: (Upper_Seq C,n) /. (k + 1) in Vertical_Line (((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2) by JORDAN6:def 6;
A43: k + 1 in dom (Upper_Seq C,n) by A16, A19, FINSEQ_3:27;
k + 1 in Seg ((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 A16, A17, FINSEQ_1:3;
then (Upper_Seq C,n) /. (k + 1) in rng (mid (Upper_Seq C,n),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 A41, A43, PARTFUN2:36;
then (Upper_Seq C,n) /. (k + 1) in {(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 A42, A35, A40, XBOOLE_0:def 4;
then (Upper_Seq C,n) /. (k + 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)) by TARSKI:def 1;
hence contradiction by A17, A12, A43, A36, PARTFUN2:17; :: thesis: verum

end;

i1 < Center (Gauge C,n) by A6, A17, A21, A30, A26, A14, A7, A8, JORDAN1A:39, NAT_1:13, NAT_1:14;
then i1 + 1 <= Center (Gauge C,n) by NAT_1:13;
then ((Upper_Seq C,n) /. (k + 1)) `1 <= ((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2 by A23, A26, A7, A8, A31, A32, JORDAN1A:39;
hence ((Upper_Seq C,n) /. (k + 1)) `1 < ((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2 by A33, XXREAL_0:1; :: thesis: verum

end;

suppose ( i1 = i2 + 1 & j1 = j2 ) ; :: thesis: ((Upper_Seq C,n) /. (k + 1)) `1 < ((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2

then i2 < i1 by NAT_1:13;
then ((Upper_Seq C,n) /. (k + 1)) `1 <= ((Upper_Seq C,n) /. k) `1 by A21, A23, A30, A26, A28, A27, JORDAN1A:39;
hence ((Upper_Seq C,n) /. (k + 1)) `1 < ((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2 by A6, A17, NAT_1:13, NAT_1:14, XXREAL_0:2; :: thesis: verum

end;

suppose ( i1 = i2 & j1 = j2 + 1 ) ; :: thesis: ((Upper_Seq C,n) /. (k + 1)) `1 < ((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2

then ((Upper_Seq C,n) /. k) `1 = ((Gauge C,n) * i2,1) `1 by A21, A25, A30, A26, GOBOARD5:3
.= ((Upper_Seq C,n) /. (k + 1)) `1 by A23, A29, A28, A27, GOBOARD5:3 ;
hence ((Upper_Seq C,n) /. (k + 1)) `1 < ((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2 by A6, A17, NAT_1:13, NAT_1:14; :: thesis: verum

end;

end;

end;

hence ((Upper_Seq C,n) /. (k + 1)) `1 < ((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2 ; :: thesis: verum

end;

A44: S₁[1]

proof

assume that
1 <= 1 and
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) ; :: thesis: ((Upper_Seq C,n) /. 1) `1 < ((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2
(Upper_Seq C,n) /. 1 = W-min (L~ (Cage C,n)) by JORDAN1F:5;
hence ((Upper_Seq C,n) /. 1) `1 < ((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2 by A3, EUCLID:56; :: thesis: verum

end;

A45: for k being non empty Nat holds S₁[k] from NAT_1:sch 10(A44, A5);
let k be Element of NAT ; :: thesis: ( 1 <= k & k < (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) implies ((Upper_Seq C,n) /. k) `1 < ((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2 )
assume ( 1 <= k & k < (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: ((Upper_Seq C,n) /. k) `1 < ((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2
hence ((Upper_Seq C,n) /. k) `1 < ((W-bound (L~ (Cage C,n))) + (E-bound (L~ (Cage C,n)))) / 2 by A45; :: thesis: verum