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 S-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 ) } = south_halfline x by TOPREAL1:41;
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 <= len (Gauge C,n) holds
(Cage C,n) /. k <> (Gauge C,n) * t,1 ; :: thesis:

A5: now

south_halfline x meets L~ (Cage C,n) by A2, Th74;
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 `2 < x `2

proof

assume q `2 >= x `2 ; :: thesis:
then q `2 = x `2 by A10, XXREAL_0:1;
then q = x by A9, TOPREAL3:11;
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 5;

now

per cases ) /. i) `2 <= ((Cage C,n) /. (i + 1)) `2 or ((Cage C,n) /. i) `2 >= ((Cage C,n) /. (i + 1)) `2 ) ;

suppose ((Cage C,n) /. i) `2 <= ((Cage C,n) /. (i + 1)) `2 ; :: thesis:

then ((Cage C,n) /. i) `2 <= q `2 by A8, A15, TOPREAL1:10;
then A16: ((Cage C,n) /. i) `2 < x `2 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_1:3;
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 11;
A21: 1 <= i2 by A19, MATRIX_1:39;
A22: i2 <= width (Gauge C,n) by A19, MATRIX_1:39;
A23: ( 1 <= i1 & i1 <= len (Gauge C,n) ) by A19, MATRIX_1:39;
x `2 = (S-min C) `2 by A1, PSCOMP_1:118
.= S-bound C by EUCLID:56
.= ((Gauge C,n) * i1,2) `2 by A23, JORDAN8:16 ;
then i2 < 1 + 1 by A16, A20, A22, A23, SPRECT_3:24;
then i2 <= 1 by NAT_1:13;
then (Cage C,n) /. i = (Gauge C,n) * i1,1 by A20, A21, XXREAL_0:1;
hence x in UBD (L~ (Cage C,n)) by A4, A11, A18, A23; :: thesis:

end;

suppose ((Cage C,n) /. i) `2 >= ((Cage C,n) /. (i + 1)) `2 ; :: thesis:

then q `2 >= ((Cage C,n) /. (i + 1)) `2 by A8, A15, TOPREAL1:10;
then A24: ((Cage C,n) /. (i + 1)) `2 < x `2 by A14, XXREAL_0:2;
A25: Cage C,n is_sequence_on Gauge C,n by JORDAN9:def 1;
A26: i + 1 >= 1 by NAT_1:11;
then i + 1 in Seg (len (Cage C,n)) by A12, FINSEQ_1:3;
then i + 1 in dom (Cage C,n) by FINSEQ_1:def 3;
then consider i1, i2 being Element of NAT such that
A27: [i1,i2] in Indices (Gauge C,n) and
A28: (Cage C,n) /. (i + 1) = (Gauge C,n) * i1,i2 by A25, GOBOARD1:def 11;
A29: 1 <= i2 by A27, MATRIX_1:39;
A30: i2 <= width (Gauge C,n) by A27, MATRIX_1:39;
A31: ( 1 <= i1 & i1 <= len (Gauge C,n) ) by A27, MATRIX_1:39;
x `2 = (S-min C) `2 by A1, PSCOMP_1:118
.= S-bound C by EUCLID:56
.= ((Gauge C,n) * i1,2) `2 by A31, JORDAN8:16 ;
then i2 < 1 + 1 by A24, A28, A30, A31, SPRECT_3:24;
then i2 <= 1 by NAT_1:13;
then (Cage C,n) /. (i + 1) = (Gauge C,n) * i1,1 by A28, A29, XXREAL_0:1;
hence x in UBD (L~ (Cage C,n)) by A4, A12, A26, A31; :: 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:28; :: thesis: