let n be Element of NAT ; :: thesis:
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis:
consider x being set such that
A1: x in W-most C by XBOOLE_0:def 1;
reconsider x = x as Point of (TOP-REAL 2) by A1;
A2: x in C by A1, XBOOLE_0:def 4;
set X = { q where q is Point of (TOP-REAL 2) : ( q `1 <= x `1 & q `2 = x `2 ) } ;
A3: { q where q is Point of (TOP-REAL 2) : ( q `1 <= x `1 & q `2 = x `2 ) } = west_halfline x by TOPREAL1:36;
then reconsider X = { q where q is Point of (TOP-REAL 2) : ( q `1 <= x `1 & q `2 = x `2 ) } as connected Subset of (TOP-REAL 2) ;
assume A4: for k, t being Element of NAT st 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) holds
(Cage (C,n)) /. k <> (Gauge (C,n)) * (1,t) ; :: thesis:

A5: now

west_halfline x meets L~ (Cage (C,n)) by A2, Th75;
then consider y being set such that
A6: y in X and
A7: y in L~ (Cage (C,n)) by A3, XBOOLE_0:3;
reconsider y = y as Point of (TOP-REAL 2) by A6;
consider q being Point of (TOP-REAL 2) such that
A8: y = q and
A9: q `1 <= x `1 and
A10: q `2 = x `2 by A6;
consider i being Element of NAT such that
A11: 1 <= i and
A12: i + 1 <= len (Cage (C,n)) and
A13: y in LSeg ((Cage (C,n)),i) by A7, SPPOL_2:13;
A14: q `1 < x `1

proof

assume q `1 >= x `1 ; :: thesis:
then q `1 = x `1 by A9, XXREAL_0:1;
then q = x by A10, TOPREAL3:6;
then x in C /\ (L~ (Cage (C,n))) by A2, A7, A8, XBOOLE_0:def 4;
then C meets L~ (Cage (C,n)) by XBOOLE_0:4;
hence contradiction by JORDAN10:5; :: thesis:

end;

A15: y in LSeg (((Cage (C,n)) /. i),((Cage (C,n)) /. (i + 1))) by A11, A12, A13, TOPREAL1:def 3;

now

per cases ) /. i) `1 <= ((Cage (C,n)) /. (i + 1)) `1 or ((Cage (C,n)) /. i) `1 >= ((Cage (C,n)) /. (i + 1)) `1 ) ;

suppose ((Cage (C,n)) /. i) `1 <= ((Cage (C,n)) /. (i + 1)) `1 ; :: thesis:

then ((Cage (C,n)) /. i) `1 <= q `1 by A8, A15, TOPREAL1:3;
then A16: ((Cage (C,n)) /. i) `1 < x `1 by A14, XXREAL_0:2;
A17: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def 1;
A18: i < len (Cage (C,n)) by A12, NAT_1:13;
then i in Seg (len (Cage (C,n))) by A11, FINSEQ_1:1;
then i in dom (Cage (C,n)) by FINSEQ_1:def 3;
then consider i1, i2 being Element of NAT such that
A19: [i1,i2] in Indices (Gauge (C,n)) and
A20: (Cage (C,n)) /. i = (Gauge (C,n)) * (i1,i2) by A17, GOBOARD1:def 9;
A21: 1 <= i2 by A19, MATRIX_1:38;
A22: i1 <= len (Gauge (C,n)) by A19, MATRIX_1:38;
A23: 1 <= i1 by A19, MATRIX_1:38;
A24: i2 <= width (Gauge (C,n)) by A19, MATRIX_1:38;
then A25: i2 <= len (Gauge (C,n)) by JORDAN8:def 1;
x `1 = (W-min C) `1 by A1, PSCOMP_1:31
.= W-bound C by EUCLID:52
.= ((Gauge (C,n)) * (2,i2)) `1 by A21, A25, JORDAN8:11 ;
then i1 < 1 + 1 by A16, A20, A21, A24, A22, SPRECT_3:13;
then i1 <= 1 by NAT_1:13;
then (Cage (C,n)) /. i = (Gauge (C,n)) * (1,i2) by A20, A23, XXREAL_0:1;
hence x in UBD (L~ (Cage (C,n))) by A4, A11, A18, A21, A24; :: thesis:

end;

suppose ((Cage (C,n)) /. i) `1 >= ((Cage (C,n)) /. (i + 1)) `1 ; :: thesis:

then q `1 >= ((Cage (C,n)) /. (i + 1)) `1 by A8, A15, TOPREAL1:3;
then A26: ((Cage (C,n)) /. (i + 1)) `1 < x `1 by A14, XXREAL_0:2;
A27: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def 1;
A28: i + 1 >= 1 by NAT_1:11;
then i + 1 in Seg (len (Cage (C,n))) by A12, FINSEQ_1:1;
then i + 1 in dom (Cage (C,n)) by FINSEQ_1:def 3;
then consider i1, i2 being Element of NAT such that
A29: [i1,i2] in Indices (Gauge (C,n)) and
A30: (Cage (C,n)) /. (i + 1) = (Gauge (C,n)) * (i1,i2) by A27, GOBOARD1:def 9;
A31: 1 <= i2 by A29, MATRIX_1:38;
A32: i1 <= len (Gauge (C,n)) by A29, MATRIX_1:38;
A33: 1 <= i1 by A29, MATRIX_1:38;
A34: i2 <= width (Gauge (C,n)) by A29, MATRIX_1:38;
then A35: i2 <= len (Gauge (C,n)) by JORDAN8:def 1;
x `1 = (W-min C) `1 by A1, PSCOMP_1:31
.= W-bound C by EUCLID:52
.= ((Gauge (C,n)) * (2,i2)) `1 by A31, A35, JORDAN8:11 ;
then i1 < 1 + 1 by A26, A30, A31, A34, A32, SPRECT_3:13;
then i1 <= 1 by NAT_1:13;
then (Cage (C,n)) /. (i + 1) = (Gauge (C,n)) * (1,i2) by A30, A33, XXREAL_0:1;
hence x in UBD (L~ (Cage (C,n))) by A4, A12, A28, A31, A34; :: thesis:

end;

end;

end;

hence x in UBD (L~ (Cage (C,n))) ; :: thesis:

end;

C c= BDD (L~ (Cage (C,n))) by JORDAN10:12;
then x in (BDD (L~ (Cage (C,n)))) /\ (UBD (L~ (Cage (C,n)))) by A2, A5, XBOOLE_0:def 4;
then BDD (L~ (Cage (C,n))) meets UBD (L~ (Cage (C,n))) by XBOOLE_0:4;
hence contradiction by JORDAN2C:24; :: thesis: