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 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 + 1),j1] in Indices (Gauge C,n) & f /. (len f) = (Gauge C,n) * (i1 + 1),j1 holds
[(i1 + 1),(j1 + 1)] 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 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 + 1),j1] in Indices (Gauge C,n) & f /. (len f) = (Gauge C,n) * (i1 + 1),j1 holds
[(i1 + 1),(j1 + 1)] in Indices (Gauge C,n)
set G = Gauge C,n;
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 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 + 1),j1] in Indices (Gauge C,n) & f /. (len f) = (Gauge C,n) * (i1 + 1),j1 holds
[(i1 + 1),(j1 + 1)] in Indices (Gauge C,n) )
assume that
A1: f is_sequence_on Gauge C,n and
A2: len f > 1 ; :: thesis: for i1, j1 being Element of NAT st 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 + 1),j1] in Indices (Gauge C,n) & f /. (len f) = (Gauge C,n) * (i1 + 1),j1 holds
[(i1 + 1),(j1 + 1)] in Indices (Gauge C,n)
A3: ( 1 <= (len f) -' 1 & ((len f) -' 1) + 1 = len f ) by A2, NAT_D:49, XREAL_1:237;
A4: len (Gauge C,n) = width (Gauge C,n) by JORDAN8:def 1;
let i1, j1 be Element of NAT ; :: thesis: ( 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 + 1),j1] in Indices (Gauge C,n) & f /. (len f) = (Gauge C,n) * (i1 + 1),j1 implies [(i1 + 1),(j1 + 1)] in Indices (Gauge C,n) )
assume that
A5: left_cell f,((len f) -' 1),(Gauge C,n) meets C and
A6: [i1,j1] in Indices (Gauge C,n) and
A7: f /. ((len f) -' 1) = (Gauge C,n) * i1,j1 and
A8: [(i1 + 1),j1] in Indices (Gauge C,n) and
A9: f /. (len f) = (Gauge C,n) * (i1 + 1),j1 ; :: thesis: [(i1 + 1),(j1 + 1)] in Indices (Gauge C,n)
A10: j1 <= width (Gauge C,n) by A8, MATRIX_1:39;
A11: i1 <= len (Gauge C,n) by A6, MATRIX_1:39;

A12: now

assume j1 + 1 > len (Gauge C,n) ; :: thesis: contradiction
then A13: (len (Gauge C,n)) + 1 <= j1 + 1 by NAT_1:13;
j1 + 1 <= (len (Gauge C,n)) + 1 by A4, A10, XREAL_1:8;
then j1 + 1 = (len (Gauge C,n)) + 1 by A13, XXREAL_0:1;
then cell (Gauge C,n),i1,(len (Gauge C,n)) meets C by A1, A5, A6, A7, A8, A9, A3, GOBRD13:24;
hence contradiction by A11, JORDAN8:18; :: thesis: verum

end;

A14: 1 <= j1 + 1 by NAT_1:11;
( 1 <= i1 + 1 & i1 + 1 <= len (Gauge C,n) ) by A8, MATRIX_1:39;
hence [(i1 + 1),(j1 + 1)] in Indices (Gauge C,n) by A4, A14, A12, MATRIX_1:37; :: thesis: verum