let n be Nat; :: thesis:
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis:
consider p being Point of (TOP-REAL 2) such that
A1: (west_halfline (W-max C)) /\ (L~ (Cage (C,n))) = {p} by JORDAN1A:89, PSCOMP_1:34;
A2: p in (west_halfline (W-max C)) /\ (L~ (Cage (C,n))) by A1, TARSKI:def 1;
then A3: p in west_halfline (W-max C) by XBOOLE_0:def 4;
A4: W-max C = |[((W-max C) `1),((W-max C) `2)]| by EUCLID:53;
A5: len (Gauge (C,n)) >= 4 by JORDAN8:10;
then A6: 1 < len (Gauge (C,n)) by XXREAL_0:2;
A7: (W-max C) `1 = W-bound C by EUCLID:52
.= ((Gauge (C,n)) * (2,1)) `1 by A6, JORDAN8:11 ;
A8: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def 1;
A9: W-max C in W-most C by PSCOMP_1:34;
p in L~ (Cage (C,n)) by A2, XBOOLE_0:def 4;
then consider i being Nat such that
A10: 1 <= i and
A11: i + 1 <= len (Cage (C,n)) and
A12: p in LSeg ((Cage (C,n)),i) by SPPOL_2:13;
take i ; :: thesis:
A13: LSeg ((Cage (C,n)),i) = LSeg (((Cage (C,n)) /. i),((Cage (C,n)) /. (i + 1))) by A10, A11, TOPREAL1:def 3;
thus A14: ( 1 <= i & i < len (Cage (C,n)) ) by A10, A11, NAT_1:13; :: thesis:
then A15: ((Cage (C,n)) /. i) `1 = p `1 by A3, A12, A9, A13, JORDAN1A:81, SPPOL_1:41;
A16: ((Cage (C,n)) /. (i + 1)) `1 = p `1 by A3, A12, A14, A9, A13, JORDAN1A:81, SPPOL_1:41;
A17: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def 1;
then consider i1, j1, i2, j2 being Nat such that
A18: [i1,j1] in Indices (Gauge (C,n)) and
A19: (Cage (C,n)) /. i = (Gauge (C,n)) * (i1,j1) and
A20: [i2,j2] in Indices (Gauge (C,n)) and
A21: (Cage (C,n)) /. (i + 1) = (Gauge (C,n)) * (i2,j2) and
A22: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A10, A11, JORDAN8:3;
A23: i1 <= len (Gauge (C,n)) by A18, MATRIX_0:32;
A24: i2 <= len (Gauge (C,n)) by A20, MATRIX_0:32;
A25: j2 <= width (Gauge (C,n)) by A20, MATRIX_0:32;
A26: j1 <= width (Gauge (C,n)) by A18, MATRIX_0:32;
A27: 1 <= j1 by A18, MATRIX_0:32;
p `1 = W-bound (L~ (Cage (C,n))) by A2, JORDAN1A:85, PSCOMP_1:34;
then ((Gauge (C,n)) * (i1,j1)) `1 = ((Gauge (C,n)) * (1,j1)) `1 by A19, A15, A8, A27, A26, JORDAN1A:73;
then A28: 1 >= i1 by A23, A27, A26, GOBOARD5:3;
A29: 1 <= i1 by A18, MATRIX_0:32;
then A30: i1 = 1 by A28, XXREAL_0:1;
A31: 1 <= i2 by A20, MATRIX_0:32;
A32: i1 = i2

proof

assume i1 <> i2 ; :: thesis:
then ( i1 < i2 or i2 < i1 ) by XXREAL_0:1;
hence contradiction by A19, A21, A22, A15, A16, A29, A23, A31, A24, A27, A25, GOBOARD5:3; :: thesis:

end;

then A33: j1 < width (Gauge (C,n)) by A10, A11, A18, A19, A20, A21, A22, A25, A28, JORDAN10:2, NAT_1:13;
j1 <= j1 + 1 by NAT_1:11;
then A34: ((Cage (C,n)) /. i) `2 <= ((Cage (C,n)) /. (i + 1)) `2 by A10, A11, A18, A19, A20, A21, A22, A29, A23, A27, A25, A32, A28, JORDAN10:2, JORDAN1A:19;
then p `2 <= ((Cage (C,n)) /. (i + 1)) `2 by A12, A13, TOPREAL1:4;
then A35: (W-max C) `2 <= ((Gauge (C,n)) * (1,(j1 + 1))) `2 by A3, A10, A11, A18, A19, A20, A21, A22, A32, A30, JORDAN10:2, TOPREAL1:def 13;
((Cage (C,n)) /. i) `2 <= p `2 by A12, A13, A34, TOPREAL1:4;
then A36: ((Gauge (C,n)) * (1,j1)) `2 <= (W-max C) `2 by A3, A19, A30, TOPREAL1:def 13;
1 + 1 <= len (Gauge (C,n)) by A5, XXREAL_0:2;
then ((Gauge (C,n)) * (i1,1)) `1 <= (W-max C) `1 by A8, A30, A6, A7, SPRECT_3:13;
then W-max C in { |[r,s]| where r, s is Real : ( ((Gauge (C,n)) * (i1,1)) `1 <= r & r <= ((Gauge (C,n)) * ((i1 + 1),1)) `1 & ((Gauge (C,n)) * (1,j1)) `2 <= s & s <= ((Gauge (C,n)) * (1,(j1 + 1))) `2 ) } by A30, A7, A36, A35, A4;
then W-max C in cell ((Gauge (C,n)),i1,j1) by A27, A30, A33, A6, GOBRD11:32;
hence W-max C in right_cell ((Cage (C,n)),i,(Gauge (C,n))) by A10, A11, A17, A18, A19, A20, A21, A22, A32, A28, GOBRD13:22, JORDAN10:2; :: thesis: