let i, n be Element of NAT ; :: thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in E-most C & p in east_halfline x & 1 <= i & i < len (Cage C,n) & p in LSeg (Cage C,n),i holds
LSeg (Cage C,n),i is vertical
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: for x, p being Point of (TOP-REAL 2) st x in E-most C & p in east_halfline x & 1 <= i & i < len (Cage C,n) & p in LSeg (Cage C,n),i holds
LSeg (Cage C,n),i is vertical
let x, p be Point of (TOP-REAL 2); :: thesis: ( x in E-most C & p in east_halfline x & 1 <= i & i < len (Cage C,n) & p in LSeg (Cage C,n),i implies LSeg (Cage C,n),i is vertical )
set G = Gauge C,n;
set f = Cage C,n;
assume that
A1: x in E-most C and
A2: p in east_halfline x and
A3: 1 <= i and
A4: i < len (Cage C,n) and
A5: p in LSeg (Cage C,n),i ; :: thesis: LSeg (Cage C,n),i is vertical
A6: i + 1 <= len (Cage C,n) by A4, NAT_1:13;
then A7: LSeg (Cage C,n),i = LSeg ((Cage C,n) /. i),((Cage C,n) /. (i + 1)) by A3, TOPREAL1:def 5;
4 <= len (Gauge C,n) by JORDAN8:13;
then A8: 1 < len (Gauge C,n) by XXREAL_0:2;
A9: len (Gauge C,n) = width (Gauge C,n) by JORDAN8:def 1;
then A10: ( 1 <= (len (Gauge C,n)) -' 1 & (len (Gauge C,n)) -' 1 <= width (Gauge C,n) ) by A8, NAT_D:35, NAT_D:49;
assume A11: not LSeg (Cage C,n),i is vertical ; :: thesis: contradiction
A12: x in C by A1, XBOOLE_0:def 4;
p in L~ (Cage C,n) by A5, SPPOL_2:17;
then A13: p in (east_halfline x) /\ (L~ (Cage C,n)) by A2, XBOOLE_0:def 4;
A14: x `2 = p `2 by A2, TOPREAL1:def 13
.= ((Cage C,n) /. (i + 1)) `2 by A5, A7, A11, SPPOL_1:41, SPPOL_1:63 ;
A15: x `2 = p `2 by A2, TOPREAL1:def 13
.= ((Cage C,n) /. i) `2 by A5, A7, A11, SPPOL_1:41, SPPOL_1:63 ;
A16: Cage C,n is_sequence_on Gauge C,n by JORDAN9:def 1;
i in Seg (len (Cage C,n)) by A3, A4, FINSEQ_1:3;
then A17: i in dom (Cage C,n) by FINSEQ_1:def 3;
1 <= i + 1 by A3, NAT_1:13;
then i + 1 in Seg (len (Cage C,n)) by A6, FINSEQ_1:3;
then A18: i + 1 in dom (Cage C,n) by FINSEQ_1:def 3;

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

suppose A19: ((Cage C,n) /. i) `1 <= ((Cage C,n) /. (i + 1)) `1 ; :: thesis: contradiction

then p `1 <= ((Cage C,n) /. (i + 1)) `1 by A5, A7, TOPREAL1:9;
then A20: ((Cage C,n) /. (i + 1)) `1 > x `1 by A12, A13, Th96, XXREAL_0:2;
consider i1, i2 being Element of NAT such that
A21: [i1,i2] in Indices (Gauge C,n) and
A22: (Cage C,n) /. (i + 1) = (Gauge C,n) * i1,i2 by A16, A18, GOBOARD1:def 11;
A23: ( 1 <= i2 & i2 <= width (Gauge C,n) & 1 <= i1 & i1 <= len (Gauge C,n) ) by A21, MATRIX_1:39;
A24: x `1 = (E-min C) `1 by A1, PSCOMP_1:108
.= E-bound C by EUCLID:56
.= ((Gauge C,n) * ((len (Gauge C,n)) -' 1),i2) `1 by A9, A23, JORDAN8:15 ;
consider j1, j2 being Element of NAT such that
A25: [j1,j2] in Indices (Gauge C,n) and
A26: (Cage C,n) /. i = (Gauge C,n) * j1,j2 by A16, A17, GOBOARD1:def 11;
A27: ( 1 <= j2 & j2 <= width (Gauge C,n) & 1 <= j1 & j1 <= len (Gauge C,n) ) by A25, MATRIX_1:39;

now

assume ((Cage C,n) /. i) `1 = ((Cage C,n) /. (i + 1)) `1 ; :: thesis: contradiction
then A28: (Cage C,n) /. i = (Cage C,n) /. (i + 1) by A14, A15, TOPREAL3:11;
then A29: ( i1 = j1 & i2 = j2 ) by A21, A22, A25, A26, GOBOARD1:21;
(abs (i1 - j1)) + (abs (i2 - j2)) = 1 by A16, A17, A18, A21, A22, A25, A26, A28, GOBOARD1:def 11;
then 1 = 0 + (abs (i2 - j2)) by A29, GOBOARD7:2
.= 0 + 0 by A29, GOBOARD7:2 ;
hence contradiction ; :: thesis: verum

end;

then ((Cage C,n) /. i) `1 < ((Cage C,n) /. (i + 1)) `1 by A19, XXREAL_0:1;
then i1 > j1 by A22, A23, A26, A27, Th39;
then len (Gauge C,n) > j1 by A23, XXREAL_0:2;
then (len (Gauge C,n)) -' 1 >= j1 by NAT_D:49;
then x `1 >= ((Cage C,n) /. i) `1 by A9, A10, A23, A24, A26, A27, Th39;
then x in L~ (Cage C,n) by A7, A14, A15, A20, GOBOARD7:9, SPPOL_2:17;
then x in (L~ (Cage C,n)) /\ C by A12, XBOOLE_0:def 4;
then L~ (Cage C,n) meets C by XBOOLE_0:4;
hence contradiction by JORDAN10:5; :: thesis: verum

end;

suppose A30: ((Cage C,n) /. i) `1 >= ((Cage C,n) /. (i + 1)) `1 ; :: thesis: contradiction

then p `1 <= ((Cage C,n) /. i) `1 by A5, A7, TOPREAL1:9;
then A31: ((Cage C,n) /. i) `1 > x `1 by A12, A13, Th96, XXREAL_0:2;
consider i1, i2 being Element of NAT such that
A32: [i1,i2] in Indices (Gauge C,n) and
A33: (Cage C,n) /. i = (Gauge C,n) * i1,i2 by A16, A17, GOBOARD1:def 11;
A34: ( 1 <= i2 & i2 <= width (Gauge C,n) & 1 <= i1 & i1 <= len (Gauge C,n) ) by A32, MATRIX_1:39;
A35: x `1 = (E-min C) `1 by A1, PSCOMP_1:108
.= E-bound C by EUCLID:56
.= ((Gauge C,n) * ((len (Gauge C,n)) -' 1),i2) `1 by A9, A34, JORDAN8:15 ;
consider j1, j2 being Element of NAT such that
A36: [j1,j2] in Indices (Gauge C,n) and
A37: (Cage C,n) /. (i + 1) = (Gauge C,n) * j1,j2 by A16, A18, GOBOARD1:def 11;
A38: ( 1 <= j2 & j2 <= width (Gauge C,n) & 1 <= j1 & j1 <= len (Gauge C,n) ) by A36, MATRIX_1:39;

now

assume ((Cage C,n) /. i) `1 = ((Cage C,n) /. (i + 1)) `1 ; :: thesis: contradiction
then A39: (Cage C,n) /. i = (Cage C,n) /. (i + 1) by A14, A15, TOPREAL3:11;
then A40: ( i1 = j1 & i2 = j2 ) by A32, A33, A36, A37, GOBOARD1:21;
(abs (j1 - i1)) + (abs (j2 - i2)) = 1 by A16, A17, A18, A32, A33, A36, A37, A39, GOBOARD1:def 11;
then 1 = 0 + (abs (i2 - j2)) by A40, GOBOARD7:2
.= 0 + 0 by A40, GOBOARD7:2 ;
hence contradiction ; :: thesis: verum

end;

then ((Cage C,n) /. (i + 1)) `1 < ((Cage C,n) /. i) `1 by A30, XXREAL_0:1;
then i1 > j1 by A33, A34, A37, A38, Th39;
then len (Gauge C,n) > j1 by A34, XXREAL_0:2;
then (len (Gauge C,n)) -' 1 >= j1 by NAT_D:49;
then x `1 >= ((Cage C,n) /. (i + 1)) `1 by A9, A10, A34, A35, A37, A38, Th39;
then x in L~ (Cage C,n) by A7, A14, A15, A31, GOBOARD7:9, SPPOL_2:17;
then x in (L~ (Cage C,n)) /\ C by A12, XBOOLE_0:def 4;
then L~ (Cage C,n) meets C by XBOOLE_0:4;
hence contradiction by JORDAN10:5; :: thesis: verum

end;

end;