let C be non empty compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: for n being 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 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,(j1 + 1)] in Indices (Gauge (C,n)) & f /. (len f) = (Gauge (C,n)) * (i1,(j1 + 1)) holds
[(i1 -' 1),(j1 + 1)] in Indices (Gauge (C,n))
let n be 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 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,(j1 + 1)] in Indices (Gauge (C,n)) & f /. (len f) = (Gauge (C,n)) * (i1,(j1 + 1)) 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 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,(j1 + 1)] in Indices (Gauge (C,n)) & f /. (len f) = (Gauge (C,n)) * (i1,(j1 + 1)) 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 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,(j1 + 1)] in Indices (Gauge (C,n)) & f /. (len f) = (Gauge (C,n)) * (i1,(j1 + 1)) 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_1:235;
let i1, j1 be 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,(j1 + 1)] in Indices (Gauge (C,n)) & f /. (len f) = (Gauge (C,n)) * (i1,(j1 + 1)) implies [(i1 -' 1),(j1 + 1)] in Indices (Gauge (C,n)) )
assume that
A4: left_cell (f,((len f) -' 1),(Gauge (C,n))) meets C and
A5: [i1,j1] in Indices (Gauge (C,n)) and
A6: f /. ((len f) -' 1) = (Gauge (C,n)) * (i1,j1) and
A7: [i1,(j1 + 1)] in Indices (Gauge (C,n)) and
A8: f /. (len f) = (Gauge (C,n)) * (i1,(j1 + 1)) ; :: thesis: [(i1 -' 1),(j1 + 1)] in Indices (Gauge (C,n))
A9: 1 <= i1 by A7, MATRIX_0:32;
A10: ( len (Gauge (C,n)) = width (Gauge (C,n)) & j1 <= width (Gauge (C,n)) ) by A5, JORDAN8:def 1, MATRIX_0:32;

A11: now :: thesis: not i1 -' 1 < 1

assume i1 -' 1 < 1 ; :: thesis: contradiction
then i1 <= 1 by NAT_1:14, NAT_D:36;
then i1 = 1 by A9, XXREAL_0:1;
then cell ((Gauge (C,n)),(1 -' 1),j1) meets C by A1, A4, A5, A6, A7, A8, A3, GOBRD13:21;
then cell ((Gauge (C,n)),0,j1) meets C by XREAL_1:232;
hence contradiction by A10, JORDAN8:18; :: thesis: verum

end;

A12: i1 -' 1 <= i1 by NAT_D:35;
i1 <= len (Gauge (C,n)) by A5, MATRIX_0:32;
then A13: i1 -' 1 <= len (Gauge (C,n)) by A12, XXREAL_0:2;
( 1 <= j1 + 1 & j1 + 1 <= width (Gauge (C,n)) ) by A7, MATRIX_0:32;
hence [(i1 -' 1),(j1 + 1)] in Indices (Gauge (C,n)) by A13, A11, MATRIX_0:30; :: thesis: verum