let n be Nat; :: thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) ex i being Nat st
( 1 <= i & i < len (Cage (C,n)) & N-max C in right_cell ((Cage (C,n)),i,(Gauge (C,n))) )
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: ex i being Nat st
( 1 <= i & i < len (Cage (C,n)) & N-max C in right_cell ((Cage (C,n)),i,(Gauge (C,n))) )
consider p being Point of (TOP-REAL 2) such that
A1: (north_halfline (N-max C)) /\ (L~ (Cage (C,n))) = {p} by JORDAN1A:86, PSCOMP_1:42;
A2: p in (north_halfline (N-max C)) /\ (L~ (Cage (C,n))) by A1, TARSKI:def 1;
then A3: p in north_halfline (N-max C) by XBOOLE_0:def 4;
p in L~ (Cage (C,n)) by A2, XBOOLE_0:def 4;
then consider i being Nat such that
A4: 1 <= i and
A5: i + 1 <= len (Cage (C,n)) and
A6: p in LSeg ((Cage (C,n)),i) by SPPOL_2:13;
take i ; :: thesis: ( 1 <= i & i < len (Cage (C,n)) & N-max C in right_cell ((Cage (C,n)),i,(Gauge (C,n))) )
A7: LSeg ((Cage (C,n)),i) = LSeg (((Cage (C,n)) /. i),((Cage (C,n)) /. (i + 1))) by A4, A5, TOPREAL1:def 3;
A8: len (Gauge (C,n)) >= 4 by JORDAN8:10;
then A9: 1 < len (Gauge (C,n)) by XXREAL_0:2;
A10: ((len (Gauge (C,n))) -' 1) + 1 = len (Gauge (C,n)) by A8, XREAL_1:235, XXREAL_0:2;
then A11: (len (Gauge (C,n))) -' 1 < len (Gauge (C,n)) by NAT_1:13;
A12: N-max C = |[((N-max C) `1),((N-max C) `2)]| by EUCLID:53;
A13: (len (Gauge (C,n))) -' 1 <= len (Gauge (C,n)) by NAT_D:44;
A14: (N-max C) `2 = N-bound C by EUCLID:52
.= ((Gauge (C,n)) * (1,((len (Gauge (C,n))) -' 1))) `2 by A9, JORDAN8:14 ;
A15: N-max C in N-most C by PSCOMP_1:42;
A16: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def 1;
thus A17: ( 1 <= i & i < len (Cage (C,n)) ) by A4, A5, NAT_1:13; :: thesis: N-max C in right_cell ((Cage (C,n)),i,(Gauge (C,n)))
then A18: ((Cage (C,n)) /. i) `2 = p `2 by A3, A6, A15, A7, JORDAN1A:78, SPPOL_1:40;
A19: ((Cage (C,n)) /. (i + 1)) `2 = p `2 by A3, A6, A17, A15, A7, JORDAN1A:78, SPPOL_1:40;
A20: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def 1;
then consider i1, j1, i2, j2 being Nat such that
A21: [i1,j1] in Indices (Gauge (C,n)) and
A22: (Cage (C,n)) /. i = (Gauge (C,n)) * (i1,j1) and
A23: [i2,j2] in Indices (Gauge (C,n)) and
A24: (Cage (C,n)) /. (i + 1) = (Gauge (C,n)) * (i2,j2) and
A25: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A4, A5, JORDAN8:3;
A26: 1 <= i1 by A21, MATRIX_0:32;
A27: j2 <= width (Gauge (C,n)) by A23, MATRIX_0:32;
A28: i1 <= len (Gauge (C,n)) by A21, MATRIX_0:32;
A29: 1 <= j1 by A21, MATRIX_0:32;
p `2 = N-bound (L~ (Cage (C,n))) by A2, JORDAN1A:82, PSCOMP_1:42;
then ((Gauge (C,n)) * (i1,j1)) `2 = ((Gauge (C,n)) * (i1,(len (Gauge (C,n))))) `2 by A22, A18, A26, A28, JORDAN1A:70;
then A30: len (Gauge (C,n)) <= j1 by A16, A26, A28, A29, GOBOARD5:4;
A31: j1 <= width (Gauge (C,n)) by A21, MATRIX_0:32;
then A32: j1 = len (Gauge (C,n)) by A16, A30, XXREAL_0:1;
A33: 1 <= j2 by A23, MATRIX_0:32;
A34: j1 = j2

proof

assume j1 <> j2 ; :: thesis: contradiction
then ( j1 < j2 or j2 < j1 ) by XXREAL_0:1;
hence contradiction by A22, A24, A25, A18, A19, A26, A28, A29, A27, A33, A31, GOBOARD5:4; :: thesis: verum

end;

A35: 1 <= i2 by A23, MATRIX_0:32;
A36: i2 <= len (Gauge (C,n)) by A23, MATRIX_0:32;
i1 <= i1 + 1 by NAT_1:11;
then A37: ((Cage (C,n)) /. i) `1 <= ((Cage (C,n)) /. (i + 1)) `1 by A4, A5, A21, A22, A23, A24, A25, A16, A26, A36, A29, A27, A34, A30, JORDAN10:4, JORDAN1A:18;
then p `1 <= ((Cage (C,n)) /. (i + 1)) `1 by A6, A7, TOPREAL1:3;
then (N-max C) `1 <= ((Gauge (C,n)) * ((i1 + 1),(len (Gauge (C,n))))) `1 by A3, A4, A5, A21, A22, A23, A24, A25, A16, A34, A32, JORDAN10:4, TOPREAL1:def 10;
then A38: (N-max C) `1 <= ((Gauge (C,n)) * ((i1 + 1),1)) `1 by A4, A5, A21, A22, A23, A24, A25, A16, A35, A36, A34, A30, A9, GOBOARD5:2, JORDAN10:4;
((Cage (C,n)) /. i) `1 <= p `1 by A6, A7, A37, TOPREAL1:3;
then ((Gauge (C,n)) * (i1,(len (Gauge (C,n))))) `1 <= (N-max C) `1 by A3, A22, A32, TOPREAL1:def 10;
then A39: ((Gauge (C,n)) * (i1,1)) `1 <= (N-max C) `1 by A16, A26, A28, A9, GOBOARD5:2;
len (Gauge (C,n)) >= 1 + 1 by A8, XXREAL_0:2;
then A40: (len (Gauge (C,n))) - 1 >= 1 by XREAL_1:19;
then (len (Gauge (C,n))) -' 1 >= 1 by XREAL_0:def 2;
then ((Gauge (C,n)) * (1,j1)) `2 >= (N-max C) `2 by A16, A32, A9, A14, A13, SPRECT_3:12;
then A41: N-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 -' 1))) `2 <= s & s <= ((Gauge (C,n)) * (1,j1)) `2 ) } by A32, A14, A39, A38, A12;
A42: 1 <= i1 by A21, MATRIX_0:32;
i1 < len (Gauge (C,n)) by A4, A5, A21, A22, A23, A24, A25, A16, A36, A34, A30, JORDAN10:4, NAT_1:13;
then N-max C in cell ((Gauge (C,n)),i1,(j1 -' 1)) by A16, A32, A42, A40, A10, A11, A41, GOBRD11:32;
hence N-max C in right_cell ((Cage (C,n)),i,(Gauge (C,n))) by A4, A5, A20, A21, A22, A23, A24, A25, A16, A34, A30, GOBRD13:24, JORDAN10:4; :: thesis: verum