let C be non empty compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: for n being Element of NAT
for f being FinSequence of (TOP-REAL 2) st f is_sequence_on Gauge C,n & len f > 1 holds
for i1, j1 being Element of NAT st front_left_cell f,((len f) -' 1),(Gauge C,n) meets C & [i1,j1] in Indices (Gauge C,n) & f /. ((len f) -' 1) = (Gauge C,n) * i1,j1 & [i1,(j1 + 1)] in Indices (Gauge C,n) & f /. (len f) = (Gauge C,n) * i1,(j1 + 1) holds
[i1,(j1 + 2)] in Indices (Gauge C,n)
let n be Element of NAT ; :: thesis: for f being FinSequence of (TOP-REAL 2) st f is_sequence_on Gauge C,n & len f > 1 holds
for i1, j1 being Element of NAT st front_left_cell f,((len f) -' 1),(Gauge C,n) meets C & [i1,j1] in Indices (Gauge C,n) & f /. ((len f) -' 1) = (Gauge C,n) * i1,j1 & [i1,(j1 + 1)] in Indices (Gauge C,n) & f /. (len f) = (Gauge C,n) * i1,(j1 + 1) holds
[i1,(j1 + 2)] in Indices (Gauge C,n)
set G = Gauge C,n;
A1: len (Gauge C,n) = width (Gauge C,n) by JORDAN8:def 1;
let f be FinSequence of (TOP-REAL 2); :: thesis: ( f is_sequence_on Gauge C,n & len f > 1 implies for i1, j1 being Element of NAT st front_left_cell f,((len f) -' 1),(Gauge C,n) meets C & [i1,j1] in Indices (Gauge C,n) & f /. ((len f) -' 1) = (Gauge C,n) * i1,j1 & [i1,(j1 + 1)] in Indices (Gauge C,n) & f /. (len f) = (Gauge C,n) * i1,(j1 + 1) holds
[i1,(j1 + 2)] in Indices (Gauge C,n) )
assume that
A2: f is_sequence_on Gauge C,n and
A3: len f > 1 ; :: thesis: for i1, j1 being Element of NAT st front_left_cell f,((len f) -' 1),(Gauge C,n) meets C & [i1,j1] in Indices (Gauge C,n) & f /. ((len f) -' 1) = (Gauge C,n) * i1,j1 & [i1,(j1 + 1)] in Indices (Gauge C,n) & f /. (len f) = (Gauge C,n) * i1,(j1 + 1) holds
[i1,(j1 + 2)] in Indices (Gauge C,n)
A4: ( 1 <= (len f) -' 1 & ((len f) -' 1) + 1 = len f ) by A3, NAT_D:49, XREAL_1:237;
let i1, j1 be Element of NAT ; :: thesis: ( front_left_cell f,((len f) -' 1),(Gauge C,n) meets C & [i1,j1] in Indices (Gauge C,n) & f /. ((len f) -' 1) = (Gauge C,n) * i1,j1 & [i1,(j1 + 1)] in Indices (Gauge C,n) & f /. (len f) = (Gauge C,n) * i1,(j1 + 1) implies [i1,(j1 + 2)] in Indices (Gauge C,n) )
assume that
A5: ( front_left_cell f,((len f) -' 1),(Gauge C,n) meets C & [i1,j1] in Indices (Gauge C,n) & f /. ((len f) -' 1) = (Gauge C,n) * i1,j1 ) and
A6: [i1,(j1 + 1)] in Indices (Gauge C,n) and
A7: f /. (len f) = (Gauge C,n) * i1,(j1 + 1) ; :: thesis: [i1,(j1 + 2)] in Indices (Gauge C,n)
A8: i1 <= len (Gauge C,n) by A6, MATRIX_1:39;
A9: i1 -' 1 <= i1 by NAT_D:35;
A10: j1 + 1 <= width (Gauge C,n) by A6, MATRIX_1:39;

A11: now

assume (j1 + 1) + 1 > len (Gauge C,n) ; :: thesis: contradiction
then A12: (len (Gauge C,n)) + 1 <= (j1 + 1) + 1 by NAT_1:13;
(j1 + 1) + 1 <= (len (Gauge C,n)) + 1 by A1, A10, XREAL_1:8;
then (j1 + 1) + 1 = (len (Gauge C,n)) + 1 by A12, XXREAL_0:1;
then cell (Gauge C,n),(i1 -' 1),(len (Gauge C,n)) meets C by A2, A5, A6, A7, A4, GOBRD13:35;
hence contradiction by A8, A9, JORDAN8:18, XXREAL_0:2; :: thesis: verum

end;

A13: 1 <= (j1 + 1) + 1 by NAT_1:12;
1 <= i1 by A6, MATRIX_1:39;
hence [i1,(j1 + 2)] in Indices (Gauge C,n) by A1, A8, A13, A11, MATRIX_1:37; :: thesis: verum